Is it possible to go the services and to fast during the High Holidays (Rosh Hashanah/ Yom Kippur) while attending BOLC?

I would contact the schoolhouse ahead of time and ask. The fasting is on you, you don't need anyone's blessing for that, no one can force you to eat, but the possibility of attending services and it interfering with POI time is definitely something you need to discuss with them ahead of time. A Chaplain can give you advice and perhaps some POC's to reach out to, but they can't give you permissions for anything, the schoolhouse is your best bet.

Reach out to the cadre of the school (shoot me a message and I can privately give you a couple names to reach out to for cadre to ask if need be) and ask them these questions directly. I do know per schoolhouse rules and requirements you can only miss so much training time. Not sure about due to religious reasons. I do know when I went to BOLC short course most sundays we were off and could attend religious ceremonies except for the Sunday we were in the field. I can not recall if religious services were offered but I am guessing they were.

You will be in the long course and not in the field until the last 3 weeks or so of BOLc so I would think fasting and religious services for most of BOLc would be no issue, but reach out to cadre to be sure.

It has been my experience that schools will attempt to make all religious accommodations as humanly possible, mission dictating. The Fasting is the easy part, really. The Sunday Services, or when ever the Judaism services are offered, your ability to attend will be dependent upon the mission. If services are after POI, there shouldn't be a problem unless there is a mandatory study hall. Regardless, just check with the Cadre and Chaplain once you arrive on station to start BOLC.