1. Yes, If that's the only way you'll be doing yoga / exercising, you can. But only until you get used to doing yoga regularly, in the sense, just to get started - This is not advisable, but if it somehow keeps you motivated, its ok to do whatever you feel like for a few days!
2. You are not doing it right - If a little bit of physical activity is making you sleepy, there is something you are definitely doing wrong. Some of the causes could be ...
-Very late dinner (beyond 10pm)
-Sleeping only after 12 in the night / Disturbed sleep
-Too much yoga / exercise at one go - Don't be in a hurry. Its a process and it will take time.
-You have too much time to sleep again, after waking up. (Once you're up, have a tight schedule until 8pm)
Try correcting the above, if they apply to you.
3. To avoid sleeping, do something else that relaxes your body. Its only your body that may need rest, not your head. Laze around and watch cartoons / browse youtube, but don't sleep!