First Q: Enlargers don't come with timers, so you must buy one separately. The Gra-Lab 300 is popular and easy to use under dim safelight due to its large dial and luminous markings, and I think is still in production.
Second Q: There are two ways of making test strips:
(1) Keeping the strip stationary, and moving a mask over it periodically to uncover more and more of the strip. When done, one end of the strip was exposed the entire time, and the other end only a short time.
(2) Exposing a patch at a time, covering the entire remainder of the strip. So each patch is an independent exposure, similar to exposing frames on film. Folks typically move the strip between exposures, so that each patch will cover the same image-area. This kind of test strip is more work, but is more useful because you can put the most important parts of the image in every patch.