The depth of your starting cut depends on several factors. Thread TPI, size of your lathe, your skill and confidence level, and the material you are cutting.
If you are cutting 4 TPI on a lathe with a 3 or 5 HP motor, then you can take a very big first cut, 0.050", 0.075" or even more. You should know what your lathe and tooling is capable of. If you are cutting 72 TPI on a lathe with a fractional HP motor, then you better start with a lot less: A LOT LESS, 0.005" may be too much.
And the material and your skill level are factored in too. If you are just starting I would try using easy to thread materials like free cutting brass or leaded steel. Start easy so you can get used to the process. Then pick up speed.
On speeds, the big problem is not cutting the thread, you can thread as fast as any other cut. It is stopping at the end of the thread without running into some other feature of the part. The time honored method on a manual lathe is to have one hand on the cross feed knob and the other on the lever for the half nuts. When you SEE that you are about to reach the end of the thread, you simultaneously disengage the half nut and QUICKLY back the cutting tool out far enough to clear the OD. I usually back out approximately one turn. This assumes that you have a functioning threading dial. If not, then you have to stop the spindle, but it will coast a bit so timing is even more critical. Some lathes have automatic cutoff devices to end threading. I actually have a manual crank handle that I can attach to the back end of the spindle so I can go very slow on some critical threading operations. It allows me to stop exactly where I want to and to do so at exactly the same spot on each successive pass.
As you may surmise from the previous paragraph, the faster the spindle speed, the more difficult the timing will be. This is why many recommend slow speeds for threading. If you have a wide clearance area at the end of thread, you can crank up the spindle speed and threading will go a lot faster. Of course, practice makes perfect and after a lot of threading experience you can use faster speeds.
In a production situation there are other ways to cut an accurate thread quickly, like a die head on a tail stock turret. Or CNC, of course.