方式 (Does it mean "model"? It says system, form or formula.)
It's not a complicated word at all, but a bit roundabout if explained precisely using the English grammar. You can replace all ～方式の with "working by —", so that ボルトアクション方式の小銃 becomes "a rifle working by bolt action", but I guess you just say "a bolt action rifle" in natural English.
軍制式小銃の系列 (Does it mean military small rifle series?)
軍: army, here means the Imperial Army
制式: a term that means, dead literally, armament catalog designation; or used as an adjective to mean "officially adopted"
であり (Does it mean "have the quality of being" in this sentence?)