Yes, I've always parked at the lot right before the bridge (off on the right). It's got a fence around it, it's cheap, and there is always someone (usually a few) people who work there in the lot. I've parked there for a few days at a time without any problems whatsoever.
I don't know about hotels near the border, I haven't stayed overnight in Reynosa before - just lots of late night trips then back to the ol USofA.
I'm sure most any hotel (as long as it isn't a top tier one) will be fine with girls coming over. Don't be afraid to ask at the hotel if your girlfriend can come over. As long as you aren't throwing a Roman Orgy and having lots of women come over, I don't see a problem. Besides, why can't an American have a Mexican girlfriend. Unless she is well known to the hotel staff or has trouble written all over her, they won't know the difference (or care). Most likely they will hit you up for another $10/night as a guest fee. I've never had this request turned down anywhere I've stayed in Mexico.