Which manufacturers use which types of tubes? Will different tubes help to give my amplifier more gain?

Most major amp manufacturers use 12AX7 and sometimes 12AT7 tubes in the preamp. Various models may use 6L6, EL34, EL84, 5881 and other output tubes.

Tubes are not really classified as "hot" or by other designations pertaining to sound. Basically, they are simple devices with well-understood functions and specifications that are designed into a circuit. Simply changing tubes in your amp will not change the sound of your amp to any great degree, as it is the overall circuit design that is most responsible for the signature tone of any particular amp. It is possible for two amps to use the exact same tubes and sound entirely different.

Having said that, there is some variation among preamp tubes. Different types have different amplification factors (gain). 12AX7's have a gain of 100X, 12AT7's are 60X, 12AY7's are 40X, and 12AU7's are 20X. They are all interchangeable and in most cases can be substituted for one another. But if you want more gain and you already have 12AX7's in your amp, you can't do much more than get a distortion pedal.

As for power tubes, Groove Tubes uses a Hardness Rating system for labeling the amount of distortion a power tube will produce. Tubes rated "10" will distort the least at a given volume, whereas tubes rated "1" will distort the most at the same volume. Tubes rated in the "4-6" area are standard on most new amps. The difference is subtle, though.