I've seen add-ons for turning a dremel rotary tool into a router or drill press, though I question the soundness of its use for large woodworking projects.
Would it be feasible to construct a rig to allow a heavier duty drill press to "flip" under the table and function as a router? Or to rotate 90° to the horizontal and operate as a lathe?
This is what the Shopsmith brand tools are known for. They sell tools in a couple of different configurations, such as the Shopsmith Mark V or the Shopsmith Mark 7. Different configurations allow you to set up the tool and use it as a variety of different shop tools, depending on what is supported by the particular model:
The pros of having such a multitool are price and shop space. The most obvious cons is the time to switch between tool configurations. Another downside is that any tool that tries to do more than one job tends not to do any of them as well as a dedicated tool. And another con is the same as the first pro: its cost. It's true that you pay less for the multitool than you would for the full set of dedicated tools. But since it can do so much, it is still fairly expensive. It might be cheaper to buy just the one or two dedicated tools you are most likely to use.
As always, there are trade-offs. Shopsmith has been selling their multitools since the 1950s. Plenty of people have used them and found them to be perfectly serviceable for their needs. If you have a small shop and are looking to maximize tooling for the space, and if you don't mind the tool changeover time, then multitools like this are something you might want to consider.
Drills operate at a much slower (factor of 10) rotational speed than routers (~28,000 rpm for routers, ~3000 rpm for drill presses; both should be slower for larger bits). It's therefore not possible (or very impractical) to put a router bit in a drill press and do anything substantial with it. You would need a way to make the drill spin much, much faster, and you can't do that with a jig.
Lathes do operate at similar rotational speeds as drill presses, and they do make attachments / products for using your drill press this way (thanks @CharlieKillian!). I would be wary of turning any very hard wood on this, though, but it should work fine for smaller or infrequent projects.
What you can also do is use your drill press as a poor-man's spindle sander, and there are products available specifically for this (and also for using your drill press as a mortising machine). There are also numerous videos online on making your own sanding drums for your drill press. Again, though, the drill press is not designed to withstand sideways forces. If you're careful to not apply too much pressure when sanding, this can work just fine and not break your drill press.
No. The spindle of a drill press is much more flimsy than that of a lathe or router. It cannot withstand sideways pressure.
A drill press has its spindle in a special construction called a "quill" that allows the spindle to move up and down. This is very useful for drilling, but the compromise is that the drill press is that it is not designed to take sideways pressures.
Drill presses can only be used for milling if the material is very soft like plastic or soft woods and the cuts are very light.
