Can I check my bike

I'm not sure, I would call the airport or visit their web site. Some flights might be different than others.

Yes, provided the aircraft you are flying on can hold it, depending on the size. Be prepared to pay an "Oversized" baggage fee, which now days isn't cheap. Check with your air carrier for specifics on size & weight limitations and fees.

I believe I have seen it done but you will need to call the airline directly... maybe a higher cost

I'm not sure I'd call the airport and ask