For your first question, if you use the money for travel expenses to visit both schools, you are morally in the clear. I had one school fly me across the Atlantic Ocean, put me up in a nice hotel for a few nights, and pay for my food. Another school is going to fly me across the US and will pay for my accommodations during my visit. Morally, our situations are equivalent. Eventually, we must both decline one school and accept another. It won't be an enjoyable experience to let them know, but it's all part of the process. They are offering travel money to attract you, but they are fully aware that paying your travel expenses will not necessarily lead to an acceptance on your part.
Regarding your second question, you can definitely visit two schools in one weekend, especially if they are only two hours apart by train. You'll be exhausted by the end, but a full day will give you enough time to speak with your POI and check out the campus. You won't be able to get a good picture of the environs, but that's difficult to do even with longer visits.
In sum, I say go for it!