Implications of using a laptop always-on as a server
If the laptop is on AC power 24/7 and also has a battery (to keep it running during power cuts), will the battery be damaged?
No, but if you want you can remove the battery and run the laptop with only AC, if you are concerned about battery wear.
Overheating is clearly a potential risk, as several people warn against. Is there any other risk to laptop hardware from running continouosly, if overheating is guarded against?
Overheating really isn't a big deal, the worst thing that can happen is that the computer forcibly shuts down when the temperatures get too high. It will not start burning. If there are no obstructions around the laptop, you'll be fine.
In sum, if I were to do this for a year or two, and then try to hand over the laptop to someone else to use as a normal machine, would it function well enough for them to do so?
Removing the battery is a good way to make sure you don't wear that out. Other than that, it depends on the temperature. If you run it with ok temperatures (no more than 50-60 degrees celsius I would say) it shouldn't matter. Make sure the LCD screen is turned off (it should be if the lid is closed).
If the laptop is on AC power 24/7 and also has a battery (to keep it running during power cuts), will the battery be damaged? I could not get a clear answer on this from the various sources I found on the net, though the information here and in this SuperUser question implies that this is not a problem except in terms of screwing up the digital calibration. However, I am not sure if this is true if the laptop is also running 24 hours a day (that information seems to pertain to laptops that are charging all the time but not necessarily running all the time).
If this is something you are trying to do in the interim to learn and develop your skills, I would say yes, keep the battery in as a backup power supply. In the real world you will have generators to kick in when the power cuts out. Might as well practice with something as apposed to nothing. I.e. How long can you hang without the wall, what do you do when the battery dies. etc.
Overheating is clearly a potential risk, as several people warn against. Is there any other risk to laptop hardware from running continouosly, if overheating is guarded against?
Overheating is a potential risk. But it is in the real world too. It is always a risk. Like i said before, if you are practicing or developing your skills. What can you do about it? Do that, if it doesn't work, think of something else.
In sum, if I were to do this for a year or two, and then try to hand over the laptop to someone else to use as a normal machine, would it function well enough for them to do so? Or should I plan on this use as a server on the assumption that it would 1) either die completely or 2) be rendered useless as a laptop after prolonged use?
It doesn't really render it useless as a normal laptop unless the modifications you end up making render it not a laptop anymore. However, just making sure you have some spare parts, keep it clean, etc. The machine will be worth the couple bucks it would be without this use in a few years. Unless it is a high end machine, it's going to be worthless anyhow. Passing it down to a family member or friend that is less fortunate, anything is worth it, selling it, not so much.
-Cheers.
I think bigbadonk420 has the best answer here. however I would just like to add on to that and mention that:
Overheating is clearly a potential risk, as several people warn against. Is there any other risk to laptop hardware from running continouosly, if overheating is guarded against?
I personally use a good, powerful cooling pad on my Dell laptop that stays on 24/7 - While not actively using the laptop, but it not being exactly idle - processing files etc... it'll rise to a max of ~57 Celsius. A cooling pad would be a good investment in that case.
-Cheers, hope I helped