Ustadh, Sometimes when I get really bored, I play for a while but I write my username as "Islam = 1 God" so that I give dawah through my name and some players realized this. Does that make my time playing an act of worship?

No, and it does not strike me as something that is done for Dawah; it could even cause problems. If you want to use that time to benefit as well, then you can play a Hadith on a loop so that you are memorizing or revising them. You could also play some lectures, listen to Q&As, and learn while you are relaxing for an hour or so playing a game (if that is something you can do). I know some brothers who told me they benefit by doing so.