https://www.cern/science/accelerators/accelerator-complex

Yes, the maximum energy that a syncrotron can accelerate a beam to is determined by its size and field strength. There are multiple rings that are used to bring beams up in energy levels before feeding to the next. Each ring has many bunches of particles circulating. So each bunch has to be going close to the same speed. You wouldn't want to do all the accelerating in one ring because it wouldn't allow nearly continuous operation.

As for two intersecting points, the collisions involve colliding two beams. So there's two different kicking/injecting points one for each direction.

https://cds.cern.ch/record/2002005/files/CERN-ACC-2015-030.pdf