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.