Common grease. The one used in most places today when you need to have something that is slippery and repels water. It is these days a Synthetic product, much better than what we had 50 years ago. Many, many brand names. Most of the time it is labeled "Hi-Temp Wheel Bearing Grease" It may include "Disk Brake" or "Drum Brake" in the name. You may want to read the label, there are a couple of exotic greases for cables and such and they will say "Not for use on bearings" or something.
This is the brand sold at most Motorcycle stores.
You can find wheel bearing grease in the smallest hardware store. I think they carry it in some quickie stores around here where I buy my morning coffee.
The traditional way to compress the felt on the throw out rod is to use a soda straw to push the rod through. This is usually the right size but if this trick doesn't work for you try dental floss. Wrap the dental floss around the felt enough to compress the felt but leave the ends trailing behind as you insert rod. Once felt is inside pull dental floss out.
Did you buy a new felt for the throw out rod? The old felt will fit better. They seldom need changing and if you have a new felt you may have to cut a mm or so off the circumference to make it fit. There are reports that felts are being cut too big by BMW's supplier. The circumference is cut at an angle. A piece is trimmed to make the felt fit a smaller rod, as it were.