Visit to a different university - is it OK to bring my girlfriend along?

It is perfectly reasonable. I have done it a couple of times and I know colleagues that frequently do it. Her presence at work/social events depends on the event and how well you know the host and the events. Often the work/social line gets blurred. I am friends with a number of my colleagues partners and my partner are friends with them and their partners. Sometimes my partner will join me for social events when we have a visitor.

That said, while it is not inappropriate, the trailing partner at academic events never has worked well for us. I have been both the leading and trailing partner. The person working is often tired at the end of the day and not really in the mood to be a tourist. We find it is a better use of our time and money to travel both as tourists and really enjoy our holidays.

We often invite people to our institute, and it's not at all unusual for them to bring their girlfriend/boyfriend/family with them. This is never a problem - we're often asking them to come a long way, and it would feel strange not to accommodate their personal ties as much as possible. I imagine other institutes are similar - certainly no-one should be surprised if you ask, and I would be surprised if it caused a problem.

I would also ask about extending the visit to cover the weekend. Sometimes institutions have bureaucratic rules that can affect their ability to pay for your flights if you do that, so it's worth making sure that's not the case here. I don't think it's an unusual thing to want to do, so again they won't be surprised if you ask, and they will probably be apologetic if it's not possible.

To complement StrongBad's answer I'll address directly a few points of your question:

Is it impolite to suggest that he book a double room, and we pay half of the price from our own pocket?

No, it's not impolite: there should be no problem for them to ask refund for just half of the hotel's invoice.

It is probable that I will be invited to some social gatherings - maybe beer after a seminar, maybe dinner.

Of course it depends on your host, but as a European, and according to my experience around Europe, for a week-long visit, there might not be many social events, probably just a dinner. You can surely tell your host that you would like to come with your girlfriend.

Would it be OK to stretch the visit for an extra weekend after the research is done

Whether it's ok or not depends on the administration of the university which refunds the flight: if it's your host's, ask them; if it's yours, ask your university administration.