Hand of Midas --- When to pick on up? When is it Useful?

Here's the criteria I use as to whether you should get one or not, though it's subjective. Just remember that HoM gives +30 attack speed and experience above normal (transmuting a centaur gives equivalent exp to an entire creep wave, which is not insignificant), which is worth some subjective amount of gold. Just to review the math:

HoM = 1900, gives +30 IAS, adds about 100 GPM (more if you transmute on the dot)
Gloves of Haste = 500, +15 IAS, so HoM is probably worth at least 1000 just for its effects alone, assuming you are a right-clicker
Extra experience = 100-200 gold (for a few transmutes)

So for 1900, you get about 1200 or so of tangible effects, leaving a 700 gold difference. This can be made up in about 6 minutes. At about 9 minutes, you make up the gold completely, as you can sell it for 50% with no net gain, but still have kept the benefits of extra exp.

So as far as when you should get one:

1) You are planning to be relatively hands-off for 6-9 minutes from the time you acquire HoM
2) You are not actively working toward a farming item (Maelstrom, Battlefury, etc)
3) You can be effective without major items for the next 6-9 minutes
4) You can't get your core items within 9 minutes anyway
5) You need to farm items, but don't actually have a great farming skillset

Just remember, there is no hard cutoff times as to when you should get a HoM. All that matters is what your timeline is in relation to the current situation. Obviously, if it's at 20 minutes and you still don't have HoM, you should probably consider a different strategy.