"I've checked into lead abatement companies and there are so few to choose from in my area (Portland, Oregon), but I'll give them a call tomorrow"
Is it possible the paint is not lead base? I'd have any questionable existing floor paint tested before obtaining contractors estimates to refurbish. Any type of HazMat cleanup can be extremely expensive in both time & money.
Re linoleum: depending on when it was manufactured, old linoleum/ tile often contains asbestos (dunno about the mastic) and usually do not present a major problem unless it is sanded. My state has enacted very stringent EPA laws and it's acceptable to cover lino with another product.
There are lead testing kits that are very inexpensive that can be purchased from most hardware/home improvement stores. The contain a reactive fluid that when in contact with lead will change a specific color.
Lead abatement is not a light matter and having worked as an abatement contractor and being licensed as an approved supervisor by the state, there are very specific regulations and procedures that will need to be observed.
Test the surface first and if there are multiple layers of paint, peel back and test the other layers as well.