Don't worry about your vision on a lucid dream, you can easily control it when you are lucid.
Yes, you should write them down to work your dream recall and to help find dream signs. I don't know if you can call them "fake" lucid dream since you achieved lucidity but lost it. That normally happens with people who aren't familiar with lucid dreaming.

About the reality checks during the day is a good idea to do them but depending on the technique they aren't fundamental, for example WILDing doesn't requires that you do RCs during the day. I suggest that you keep doing RCs during the day to figure out which techniques works best for you.
But the RCs when waking up are very important to get LDs from false awaknings