I've come across a lot of trap doors and swinging blades, and since I have a Robber, I want to disarm them. But I don't know how, or even if I can. How does one disarm a trap?

He will attempt to do it automatically for trapped items and chest. For the swinging blades and such, those you have to walk through and survive.

Not to my knowledge, try using the Jump spell to get past them.
The manual doesnt mention anything about disarming traps so you heroes may not even disarm trapped chest, its just something I assumed. I will have to pay more attention to the clairvoyance spell and see if any of them are being disarmed.