The rule of onus probandi states that the burden of proof lies with the speaker, not with the one who negates, see the OP's question.

1. The three other principles mentioned in the OP’s question are characterized by wikipedia as follows:

   "Russell's teapot is an analogy, formulated by the philosopher Bertrand Russell (1872–1970), to illustrate that the philosophic burden of proof lies upon a person making empirically unfalsifiable claims, rather than shifting the burden of disproof to others”, see https://en.wikipedia.org/wiki/Russell%27s_teapot

   “Hitchens's razor is an epistemological razor that serves as a general rule for rejecting certain knowledge claims. It states "what can be asserted without evidence can also be dismissed without evidence."”, see https://en.wikipedia.org/wiki/Hitchens%27s_razor

   “Occam's razor [...] is the problem-solving principle that recommends searching for explanations constructed with the smallest possible set of elements”, see https://en.wikipedia.org/wiki/Occam%27s_razor

   All of the three principles are not laws of nature. They cannot be detected by research. Instead, they are principles of debate as @KristianBerry points out. Therefore they have to be negotiated.

2. The principles of Russell’s teapot and Hitchens’ razor do not imply that it is always the person, who makes a claim of existence (proponent), who has the burden of proof (rule of onus probandi), but not his opponent.

3. The rule of onus probandi breaks the symmetry between the proponent and the opponent, because it shifts the burden of proof to the proponent.

   A fair debate should provide some argument, why to follow this rule, and should secure the agreement of the interlocutors to accept the rule of onus probandi.

   An argument for accepting the rule of onus probandi in a debate between theism and atheism could be: The proponent can prove his claim of existence just by presenting the one entity. While the opponent in general cannot show the non-existence of the entity on all locations.

   The asymmetry is between showing the existence on one location and showing the absence on all locations. The only chance of the opponent is to detect a logical contradiction in the argument of the proponent.

   Hence, the evidence of logical inconsistency in the claim of the theist is one of the strongest arguments in the hand of the atheist. See Leibniz’ attempt to argue against the criticism uttered against the theistic position.

4. If there is agreement that the premisses of Occam razor are fullfilled, then it seems reasonable to prefer the more simple theory: Why make it complicated when you can make it simple?

The rule of onus probandi states that the burden of proof lies with the speaker, not with the one who negates, see the OP's question.

1. The three other principles mentioned in the OP’s question are characterized by wikipedia as follows:

   "Russell's teapot is an analogy, formulated by the philosopher Bertrand Russell (1872–1970), to illustrate that the philosophic burden of proof lies upon a person making empirically unfalsifiable claims, rather than shifting the burden of disproof to others”, see https://en.wikipedia.org/wiki/Russell%27s_teapot

   “Hitchens's razor is an epistemological razor that serves as a general rule for rejecting certain knowledge claims. It states "what can be asserted without evidence can also be dismissed without evidence."”, see https://en.wikipedia.org/wiki/Hitchens%27s_razor

   “Occam's razor [...] is the problem-solving principle that recommends searching for explanations constructed with the smallest possible set of elements”, see https://en.wikipedia.org/wiki/Occam%27s_razor

   All of the three principles are not laws of nature. They cannot be detected by research. Instead, they are principles of debate as @KristianBerry points out. Therefore they have to be negotiated.

2. The principles of Russell’s teapot and Hitchens’ razor do not imply that it is always the person, who makes a claim of existence (proponent), who has the burden of proof (rule of onus probandi), but not his opponent.

3. The rule of onus probandi breaks the symmetry between the proponent and the opponent, because it shifts the burden of proof to the proponent.

   A fair debate should provide some argument, why to follow this rule, and should secure the agreement of the interlocutors to accept the rule of onus probandi.

   An argument for accepting the rule of onus probandi in a debate between theism and atheism could be: The proponent can prove his claim of existence just by presenting the one entity. While the opponent in general cannot show the non-existence of the entity on all locations.

   The asymmetry is between showing the existence on one location and showing the absence on all locations. The only chance of the opponent is to detect a logical contradiction in the argument of the proponent.

   Hence, the evidence of logical inconsistency in the claim of the theist is one of the strongest arguments in the hand of the atheist. See Leibniz’ attempt to argue against the criticism uttered against the theistic position.

4. If there is agreement that the premisses of Occam razor are fullfilled, then it seems reasonable to prefer the more simple theory: Why make it complicated when you can make it simple?

1. First I would like to recall the three principles mentioned in the OP’s question:

   "Russell's teapot is an analogy, formulated by the philosopher Bertrand Russell (1872–1970), to illustrate that the philosophic burden of proof lies upon a person making empirically unfalsifiable claims, rather than shifting the burden of disproof to others”, see https://en.wikipedia.org/wiki/Russell%27s_teapot

   “Hitchens's razor is an epistemological razor that serves as a general rule for rejecting certain knowledge claims. It states "what can be asserted without evidence can also be dismissed without evidence."”, see https://en.wikipedia.org/wiki/Hitchens%27s_razor

   “Occam's razor (also spelled Ockham's razor or Ocham's razor; Latin: novacula Occami) is the problem-solving principle that recommends searching for explanations constructed with the smallest possible set of elements”, see https://en.wikipedia.org/wiki/Occam%27s_razor

   All of the three principles are not laws of nature. They cannot be detected by research. Instead, they are principles of debate as @KristianBerry points out. Therefore they have to be negotiated.

2. The principles of Russell’s teapot and Hitchens’ razor do not imply that it is always the person, who makes a claim of existence (proponent), who has the burden of proof (rule of onus probandi), but not his opponent.

3. The rule of onus probandi breaks the symmetry between the proponent and the opponent, because it shifts the burden of proof to the proponent.

   A fair debate should provide some argument, why to follow this rule, and should secure the agreement of the interlocutors to accept the rule of onus probandi.

   An argument for accepting the rule of onus probandi in a debate between theism and atheism could be: The proponent can prove his claim of existence just by presenting the one entity. While the opponent in general cannot show the non-existence of the entity on all locations.

   The asymmetry is between showing the existence on one location and showing the absence on all locations. The only chance of the opponent is to detect a logical contradiction in the argument of the proponent.

   Hence, the evidence of logical inconsistency in the claim of the theist is one of the strongest arguments in the hand of the atheist. See Leibniz’ attempt to argue against the criticism uttered against the theistic position.

4. If there is agreement that the premisses of Occam razor are fullfilled, then it seems reasonable to prefer the more simple theory: Why make it complicated when you can make it simple?

The rule of onus probandi states that the burden of proof lies with the speaker, not with the one who negates, see the OP's question.

1. The three other principles mentioned in the OP’s question are characterized by wikipedia as follows:

   "Russell's teapot is an analogy, formulated by the philosopher Bertrand Russell (1872–1970), to illustrate that the philosophic burden of proof lies upon a person making empirically unfalsifiable claims, rather than shifting the burden of disproof to others”, see https://en.wikipedia.org/wiki/Russell%27s_teapot

   “Hitchens's razor is an epistemological razor that serves as a general rule for rejecting certain knowledge claims. It states "what can be asserted without evidence can also be dismissed without evidence."”, see https://en.wikipedia.org/wiki/Hitchens%27s_razor

   “Occam's razor [...] is the problem-solving principle that recommends searching for explanations constructed with the smallest possible set of elements”, see https://en.wikipedia.org/wiki/Occam%27s_razor

   All of the three principles are not laws of nature. They cannot be detected by research. Instead, they are principles of debate as @KristianBerry points out. Therefore they have to be negotiated.

2. The principles of Russell’s teapot and Hitchens’ razor do not imply that it is always the person, who makes a claim of existence (proponent), who has the burden of proof (rule of onus probandi), but not his opponent.

3. The rule of onus probandi breaks the symmetry between the proponent and the opponent, because it shifts the burden of proof to the proponent.

   A fair debate should provide some argument, why to follow this rule, and should secure the agreement of the interlocutors to accept the rule of onus probandi.

   An argument for accepting the rule of onus probandi in a debate between theism and atheism could be: The proponent can prove his claim of existence just by presenting the one entity. While the opponent in general cannot show the non-existence of the entity on all locations.

   The asymmetry is between showing the existence on one location and showing the absence on all locations. The only chance of the opponent is to detect a logical contradiction in the argument of the proponent.

   Hence, the evidence of logical inconsistency in the claim of the theist is one of the strongest arguments in the hand of the atheist. See Leibniz’ attempt to argue against the criticism uttered against the theistic position.

4. If there is agreement that the premisses of Occam razor are fullfilled, then it seems reasonable to prefer the more simple theory: Why make it complicated when you can make it simple?