You can mix brands of transmitters and receivers. Just make certain that you have the proper connectors on your servos for the brand of receiver that you use.
The second part of the first question and the second question are very related. First, make certain that the transmitter and receiver are of the same modulation type (AM or FM). Next, make certain that the crystals in both are a match in the frequency.
As for the brand specificity of crystals, you will find that there are two schools on this. The first school (the one I belong to) holds that the brand does not matter. I recommend matching the brands when it comes to crystals though. What I mean, pair your crystals from the same manufacturer. It does not matter if the brand is the same as the radio or recevier.
The other school of thought is that brand does matter and that everything should match. My only fault with this logic is that radio waves are radio waves. If you get a frequency match along with modulation type match, you should be good to go with anything.
The reason I recommend getting crystal pairs from the same manufacturer has to do with a better frequency match from the same manufacturer. In reality, all things being equal, the crystal from one manufacturer should perform the sameway one from another would.
Do not ask me for a brand to recommend. I do not use crystals (full synth on all of my rides), and I would be a poor choice for giving you an accurate suggestion.