Can you warm biga in a microwave?

Yes, but the trick is not to have the dough in the microwave when it's on:

• place the dough into a wide, shallow bowl, and wrap it tightly with plastic wrap.
• Fill a microwave-safe bowl with water, and float a toothpick or a grain of rice in it.
• Microwave the water 'til it's boiling.
• Place the bowl with the dough above the boiling water; the water level should not touch the upper bowl.
• Close the microwave and wait.

(the plastic wrap isn't absolutely necessary ... but it prevents you from steaming the outside of the dough, changing its moisture content)

The toothpick (or similar) is necessary, as it prevents the water from heating above its boiling point and spontaneously boiling when you nudge it..

Impatience is very hard to reconcile with baking bread (or any other fermentation process, for that matter).

If it works, it still won't be a good choice. Yeast doesn't like sudden temperature shifts, the gentler the change, the better. So, the warm room will yield the most flavorful bread, and have the least chance of failure. Putting the dough somewhere slightly warmer than the room (in front of a radiator, or on top of a running dishwasher, for example) still works, although the fermentation might go a tiny bit off. You might or might not notice a difference in taste.

But the microwave is terrible at heating, and this will affect your dough a lot. Even if the yeast does not die (before reading Elendil's answer, I'd have thought this the more probable outcome), it will go through an unpleasant shock, changing its metabolism and producing who knows what - thiols and ammonia are two things that are typical in yeast colonies under stress. The microwave also heats in a very uneven way, so you will end up with unequally heated biga, another problem. And, if you are not slow and careful enough, you might give it just a bit too much juice - and there is no way to notice it when it is too much - and vaporise the cytoplasm out of your poor yeast cells, leading to dead dough and no rise in the oven, so total failure.

Frankly, using a cold biga might be the better solution when compared to the fussiness, unreliability and quality loss from microwave heating. Or just make a standard bread without biga.

Yes, you can, but you have to take a great deal of care. Use the lowest power setting on your microwave and use short bursts of power rather than nuking it.

Source: I happened to see someone do exactly this on TV the other day, and it worked, to my surprise.

Our over-the-range microwave warms up nicely when the hood light (underneath) is turned on. This may still get too warm but requires monitoring at 20 minute intervals, where the microwave heating/over-heating happens in seconds. The light in the conventional baking oven will provide the same gentle heat. Either way I would try to keep the temp under 95F although I have seen others go higher.