You ask:
"My first concern is if it's a good idea to keep the catalog on the drive, so that I have the same catalog when I use my second computer. Is there a risk with this and will this affect the speed using lightroom?"
There is no risk involved, but it would be advisable to have a backup of your catalog on your PC in case your laptop goes AWOL during travel.
"If I keep the catalog local, can I then work work (rate, add keywords etc) with the pictures without my disc connected (when traveling)?"
Without connecting the hard drive to your laptop you cannot work on your images in the Develop Module. Whenever the external drive is disconnected you will get the message that "file is offline or missing".
You will get that message on the PC when the drive is disconnected and you will get it on the laptop when the drive is not connected. It doesn't matter where the catalog is. Lr needs both: the catalog and the image files.
But for working in the Library Module you would not need the image files. So you could add keywords, titles, captions, etc.
"Will the work be much slower compared to multiple internal 7200 discs as on my stationary pc?"
Possibly yes, I have no personal experience with USB3, so can't say for sure but I think some loss of Lr performance will happen. Lr constantly writes to and reads from the catalog. So speed of drive where the catalog is located is essential to the performance of Lr.