You do have to pay to bring your bike on the train.
Are you talking Amtrak? Carry on bikes are permitted on trains with racks, first come, first served. When the racks are full, no additional bikes are permitted on train. The Pacific Surfliner has 6 racks per train. Bikes can also be checked as baggage at a $10 fee (current price subject to change) if inside a bike box. Most stations will sell you a box for an additional $15, but must be somewhat disassembled and at least one hour before departure.
Yes
Yes they but no gas powered cycles
On metrolink don't know about Amtrak.
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