The city is great too with a long standing digital background: ARM, MS Research etc.
It's definitely worth checking out, and good luck.
Points worth noting include a local Google research/dev. office (does Chrome's V8), and the "birth university" of Bjarne Stroustrup of C++ fame.
The department has research groups in the following areas:
Algorithms and Data Structures, Bioinformatics, Mathematical Computer Science, Computer Graphics and Scientific Computing, Cryptography and Security, Data-Intensive Systems, Human Computer Interaction, Modelling and Validation of Distributed Systems, Object-Oriented Software Systems, and Programming Languages
Feel free to contact me if there's anything I can help you with :)
They do have a good deal of exchange programs. But if you're not applying through one of the big networks (ERASMUS, etc.), then it's probably down to the individual university.
There are tons of exchange students from all over the world, both Master's and PhDs, so it shouldn't be a problem at all.
Edinburgh is ridiculously picturesque and historical but it does have a reputation for being, when compared to Glasgow, of not nearly such a welcoming place.
CS is quite broad and it might be wise to choose a university that has a strong research tradition in the field(s) that you're interested in.