Don't you think that after a 'hit' or two, the Christians would be complaining to the Roman authorities who would then have a word or two with the 'hitmen'?
I imagine they did. There are indications of that but not all that many.
Pax Romana was meant to gaurantee protection under Roman rule, I could see a few hits but if Jews were regularly lynching Christians I would think that sooner or later the authorities would do something about it.
Yes and no. Pax Romana was very careful not to step into local and parochial issues. You can see this in the stories of Christ's trial and crucifixion. The state did not want to get involved in intra-sect issues. Until the number of Christians reached some key level (probably around the time of Constantine), they were seen as just a different Jewish Sect and things like hits were simply inhouse squabbles. In addition, unless the folk happened to be citizens (not all that usual) they had little say anyway.
Aslan is not a Tame Lion