If my girlfriend was going to travel with me to, say, Jordan, or where-ever, should she probably wear a hijab when she's out and about in public?
In most Muslim countries, keeping the arms and legs (and of course the cleavage) covered would be considered completely sufficient, especially for someone who is visibly a foreigner.
Some (e.g. Turkey, at least the bigger cities) are much more tolerant and nobody will feel offended by bare arms or legs. The strictest dress code exists in Saudi Arabia, where women are required to wear the black abaya, but not (at least in theory) to cover their heads. However, as an unmarried couple you'd be breaking the law just by spending time together, so you'll probably not want to go there anyway.
Definitely not in Jordan, except in places like a mosque where you may want to show some respect but it would be enough to put a light foulard on your head. However, other countries may be more strict about this.
It depends on the country. On the flight to Tehran, the crew members reminded female passengers that headscarves are required attire in public, and that all women should put on a headscarf now, before the aircraft lands.
Clothing expectations are generally cultural and legal rather than religious. The best bet is to do some research for the country (or countries) you plan to visit.
For example, in Indonesia (the world's largest Muslim country by population), it is acceptable for women to dress casually in most places, bikinis on the beach, and so on. But if you're going to visit a mosque, you would probably be expected to dress more modestly.
At the other end of the spectrum, Saudi Arabia is probably the most conservative.
There is no standard for "Muslim dress" any more than there is for "Christian dress" or "Jewish dress", do some research on the country you plan to visit and act accordingly. The Lonely Planet guide books (and website) are a great resource for independent travellers.