Jurisdiction: england-and-wales
Preliminary issues
Firstly, as some of the comments have highlighted, this could be a scam. I have personally come across such a scam on two occassions. The scammer rents a property short term (e.g. 2-3 weeks) on AirBnB. They then pose as a landlord or letting agent and advertise the same property as a long term let. They collect a deposit and rent from any person who wants to be a tenant. They may even copy the keys and give each "tenant" a set. On move in day, you arrive at the property to find that you are not the only person trying to move boxes in.
It has always struck me that landlords are generally very careful to vet their tenants by checking ID, proof of address, obtaining references, and running credit checks. Yet tenants rarely do any vetting at all of their landlords.
A very basic and easy check you can do is to purchase the title register (not title plan) for the property from the Land Registry. This costs £3 and will give you the name of the person who owns the property. If the property is an apartment then you will generally want the leasehold title register (not the freehold). Once you have the name, you can then ask your landlord to provide proof that they are that person.
Secondly, you've tagged the question united-kingdom, but the UK is actually comprised of multiple legal jurisdictions and housing law varies among them (particularly in Scotland). I'm answering this on the basis of england-and-wales.
Third, questions asking for legal advice on real situations are off-topic here. My answer will just address the general issues and shouldn't be taken as advice for your situation.
Contract and due dates
There is nothing in contract law which prevents obligations from arising before the date that the contract is agreed. It is not unusual for parties to draft contracts which govern past behaviour. In that sense, it is perfectly acceptable to agree a contract on 12 August which requires rent to have been paid on 8 August (albeit it would be inadvisable to agree such a contract as you would immediately be in breach if you had not already paid). On the other hand, a contract which purports to have been agreed on a date which is earlier than when it was actually agreed, can amount to fraud.
I would be wary of a subsequent email which purports to allow a later due date which contradicts the contract. Unless there is a clause in the contract allowing for the landlord to postpone due dates, the email is unenforceable and your real due date is still 8 August. The attempt to change the due date is effectively a variation of the contract, and a variation which is not permitted in the contract itself needs to be executed as a second contract. That means you need all the elements of a contract: offer/acceptance, intention to be bound, and consideration. The problem here is the latter. The landlord is providing consideration (a later due date) but you are offering nothing in return.
Holding the room
"The agent told me they cannot hold my room too long due to the high volume of interest in booking the rooms."
"Once you have signed this agreement you will be liable for the full rent set out in the agreement unless released from your tenancy by the Landlord or Management Company."
These two positions are contradictory. If you agreed a tenancy (as implied by the second quote), then you have a contract which is legally binding on both parties. The first quote is incorrect - there is nothing to "hold" because the room is already yours.
On the other hand, it may be that what you agreed was a holding deposit agreement (rather than a tenancy agreement) which is merely a commitment on the part of the tenant to forfeit a sum of money (which by law cannot exceed 1 week's rent) in the event that the tenant (as opposed to the landlord) decides not to proceed with the tenancy.
The wording from the second quote (liability for full rent) implies that it is a tenancy agreement rather than a holding deposit agreement. Or, in the alternative, that it is an illegal holding deposit agreement which asks for more money to be forfeited than is lawful.
You'll need to read your full contract to understand what it is. If it's a tenancy agreement, it will be obvious from the wording that you have actually rented the property.
Obligation to pay rent
"My question is do I have any legal responsibility to pay for the entire rent by the new due date (25 August) for APT1?"
Unfortunately, if you have signed a tenancy agreement, then you are legally bound to fulfill your obligations under it (provided such obligations do not break the law). If it contains a clause stating that you must pay 51 weeks's rent, then that is what you must do.
"If they told me they cannot hold my booking any longer, do I need them to confirm in writing that I'm released from the agreement?"
As a general rule of contract law, nothing needs to be in writing unless (a) the law requires it to be in writing or (b) the contract requires it to be in writing. You agree non-written contracts all the time when you go shopping, use the bus, etc. The same applies to taking actions which are governed by a pre-existing contract e.g. giving your taxi driver verbal directions once you are en-route.
The phrase "unless released from your tenancy by the Landlord or Management Company" says nothing about the release needing to be in writing; therefore it can be verbal (provided that there isn't another clause somewhere else in the contract which requires it to be in writing). Be aware however that verbal statements can be difficult to prove.
