First of all if you are new to lucid dreaming don't do WILD. Do an easier technique such as MILD or DILD. If you do a WILD, you need to sleep 4.5-6 hours first. This will put your body in the optimal state for WILD and you should be set for your longest REM period of the night. Your question is kind of long so a bit hard to follow, I would break them up into multiple questions in the future. WILDs take practice and anything distracting you such as a mosquito will keep you from relaxing your mind enough to induce a WILD.

There are tons of WILD tutorials that go into more detail. If you simply use MILD everytime you wake up you will find this much easier then trying to do a WILD and less work. WILD is great once you get the hand of things, otherwise it is the hardest technique to master. The key is relaxed mind and body. If you do it after waking up your body will already be relaxed as long as you don't move at all upon awakening.

You can't control when you go into REM, although when you nap you typically will enter REM immediately. Even if you do enter REM immediately when going to bed at night (i do sometimes, but biologically it will usually be an hour or more beforehand), this is your shortest REM period (only 10-15 minutes), so you are better to shoot for the longest (45 minss+) which is your last rem cycle.