あふれかえってて is the casual contraction of 溢れ返っていて — a compound verb (溢れる 'overflow' + 返る 'return/back') intensifying the overflow into 'absolutely teeming.' 押し分けて 'pushing apart' chains into 入る for the compound motion. 気もせず uses も for emphasis ('didn't even feel like it') with the classical negative conjunctive ず. そのまま 'as-is' is a set adverb for doing nothing and leaving things unchanged.