You are allowed to take your bike on the trains - this is true since August of this year.
1 - you should arrive 20 minutes before the train leaves
2 - you need to board a specially marked train car
3 - you will need to show a lock or a dock and actually lock or dock the bike in the designated train car
4 - you may have to wait for the next train if the train is packed with passengers and/or if there are already 5 bike riders in the compartment
5 - The permitted travel times are:
Sun-Thursday 9 am - 3 pm and again from 7 pm until end of service
Fridays - all day long
Saturday evenings you may not travel with your bike
I can't fathom how you could travel with your bike on a Sherut - they are minivans, and there would be no room. I've also never seen anyone travel with their bike on the bus. But hopefully the new train policy will be sufficient for you travel plans.
I don't think you'll have any problems at checkpoints.
Your main concern is the aggressive drivers in the region, who may show very little respect for bike riders, and the infrastructure which is rarely accommodating to bike riders - you need to take extreme caution. Every year there are reported accidents, and serious bikers in Israel will generally ride with an accompanying vehicle for safety.