I am sorry to read you are STILL having problems!!
The protruding primer can be a couple things. But knowing the ignition problems, my thinking is, yes the chamber could be reamed a bit deeply.
Did you remove the barrel form the gun. Then remove the extractor and insert a FL resized case or factory loaded bullet. It should be flush or every so slightly below flush. Below flush as a correctly seated primer will sit in the primer pocket. IF its more than that then YES its excessive.
Is the primers sticking out upon firing primers only or live ammo. If just popping primers then it could be nothing. If live ammo, it still could be nothing if the loadings are very light. In these light loads the case cannot adhear to the chamber walls properly and it allows the primers to pop out a bit.
NOW in a perfect world, the primers should not "pop" reguardless of the above senerio.
You could also go to a gunsmith and have the action checked with "go/no go" gauges. If your gun closes on a "no go" it has excessive head space.
This is a new gun and H&R / Remington should make it right if in fact the chamber has been cut too deeply.
Again get Brian involved and let him know about what you have found and what you think.
CW