Many op-amps, not all but some have diodes between the inputs internally.. This will prevent you having differential ranges more than you have seen, obviously when directly driving the input.
It's obvious that Op-amp was meant to be used as a - feed back amplifier. However, all is not lost though, with R's on the input you still can use it for circuits like comparators. It'll simply clamp the diff voltage to not exceed the one diode drop.
As for the output pulling down, that sounds about normal for a monolithic type with complemary pnp/npn emitter totems on the output.
You know, there is nothing stopping you from using some old time components to form a differential from end with a couple of npn's for example, and drive the output stage. I've done it many times when making HV amplifier circuits. You may not have the gain but in most cases you don't need the gain capability of an op-amp.