1) Cheap way to fix some minor timining issue, like spurious spikes, under/overshoots or reflections on transitions. Usually you'd connect it to GND but +5V in such system is also pretty low impedance, so it could be because the +5V pin was easier to reach or a deliberate choice because it somehow helped a bit more (this depends on what was the original issue with the signal).
2) Is it really damaged? I can't tell. Looks a bit dirty from flux or glue - if it's not cracked or badly chipped it might be OK. Do check it for short though. As far as I can tell it was going to T/C but since it's desoldered perhaps the mod was undone because it was not needed after all, but without removing the now glued cap and risking ripping the solder mask off (or worse, the copper too).
3) Such filters would be somewhere between 10pF and 100pF. Lower than 10 is unlikely as the copper itself will have more capacitance so it would have to be super marginal issue to need such a small fixing cap. More than 100pF doesn't really do anything except load the signal line a lot, which is dangerous for the chips - not impossible I guess if the issue was bad and this was the only way to sell the mobo, so something like 1nF or even above is not out of question, but in that case there should be a pretty obvious difference with the cap in circuit and without it. I would expect a mobo needing a big cap here to be pretty unstable without the fix.

Note that the T/C line is not used all that much since the DMA is not something that usually runs all the time. This would mostly affect floppy R/W and cards like SoundBlaster.

The motherboard did come with a WD37C65 Floppy Disk controller card (Although there is also a 40-pin header on it so I guess it also controls hard drives?) which could be the reason for the mod I guess?