Generally, since the railroad was there first and the grade crossing is an easement on railroad right of way, any grade crossing structure/improvement is the responsiblity of whatever agency owns the road, in this case, the county. So a grade separation, be it an overhead or a tunnel, would be on the county's dime (although probably state and federal grants would be available for some of the expense).
I don't know about maintenance expenses for grade crossing signals. I think the railroad just takes care of routine maintenance as part of signal maintenance, but any capital expense, like replacing the signals, would be on the agency's dime. Not doing routine maintenance as part of a signal maintainer's duties doesn't make sense and would lead to massive tie ups since a non-functional grade crossing signal has to be flagged across, so it just makes sense to be able to dispatch a signal maintainer (which is what they do). Parsing out any grade crossing signal maintenance to the local agencies sounds like a nightmare, parsing any time and equipment to individual crossings, then billing it. Crossing 1 to Y City, Crossing 2 to X County, Crossing 3 to the state highway department, crossing 4 to Z city.