Is it legal for my city to take street parking in front of my house away for a bike trail when I only have a shared driveway with next door neighbor?

Yes, the City owns the street and can do anything reasonable it wants to. It could have painted the curb red so you could not park there. Some bike trials are placed several feet away from the curb so cars can still park without interfering with the bike lane. The city has to weigh your interests versus the safety of bicyclists.

To the extent that the right-of-way extends beyond the roadway - off of each side, typically in Michigan 60 feet total with a 30-foot road, they can widen the road without encroaching on your property.

It's public right of way, not your parking lot, so yes.