Welcome to 3rd!
1. I give my students a homework sheet every week - it's on the back of their weekly newsletter. I write the homework on the board and they write it on their homework sheet. Before I stamp their behavior log (at the bottom of their homework sheet) I check to make sure they've written down the homework.
2. We go over the homework together in class. It's the quickest way to find out if I need to review a concept or go into more detail. It takes just a few minutes when we're starting that subject. What I do is ask who doesn't have their homework - then I write their name in a binder. Then we go over it together (I don't worry about them lying about it - they know I'll walk around and see who has it). When I do graded paper folders (every Fri.) I can glance in the binder to see who didn't do homework that week. I also give out homework passes if they do their homework all month.
3. The ones who didn't do homework miss recess. They do the homework in a recess room (we take turns with that duty).
4. No, I don't normally record a grade for homework. I consider it practice, and I just want them to try. I've had problems in the past with parents doing it for them, so that really wasn't THEIR grade anyway!
5. I don't give a lot of homework - we work really hard throughout the day. I usually give a math sheet and occasionally some spelling (we don't do spelling every week). They also are required to do a reading log every month - but I count that as a project grade since it lasts the whole month.
I don't blame you for asking a lot of questions - I did the same thing!