No. In the blood test for Lyme disease one looks in the routine diagnostics for specific antibodies against the triggering bacteria, ie against the Borrelia. However, if antibodies are found in the blood, this is no sure proof that you really have Lyme disease (Lyme disease).
This question has not yet been finally clarified. The causative Borrelia, a bacterial species, have been detected in a variety of stinging insects, not just ticks. In the case of ticks, they are dormant in the intestine and are then transmitted during the blood meal.
No. The removal of the tick is harmless to the person who removes it. You can not get a Lyme disease. This is usually true even if you touch the tick while doing so (if you use a pair of tongs, you can even completely avoid it). An exception is described below.