I teach fourth grade in a big city school, though we're not inner city (we're between 55-60 percent FRL). I have a difficult classroom mix this year, as well. I took me weeks to really have my kids to a point where I was happy with their behavior.
With some difficult kids, if you build a relationship with them, you can get them to behave because you ask them to.
Student engagement is a major key to keeping kids behaving in elementary school. It's true in fourth, but even more so with the younger kids. Have you tried teaching with songs? I have my kids answer a lot of things with thumbs up/down so everyone has to participate. For short answers, sometimes I'll have the kids all call out the answer together on three or such. Call and response works well, too, especially if you add in silly voices or repeating at a whisper/loudly, etc.
If you haven't already, set up a quiet/attention signal with the class. I personally like call and response ones. Really, pretty similar to what I use at camp. In my classroom, I often use either saying 'Peace' with the class responding 'Quiet' or 'One, Two, Three eyes on me'/'One, two eyes on you.
Setting up a system with the paper slips sounds great. Have you considered having them earn the right to special time with you? Kids /love/ teacher attention. Getting to sit with you and eat lunch talking about silly kid things or play a special game with you could be good incentive for kids to get attention the right way. Positive letters/notes home are awesome as well or bragging about kids to their parents if you see parents during pick up time.
I'd ask the teacher about why she doesn't like parent contact. It might have been that particular student. I've had a couple I didn't call home to because I knew that the parent reaction would be detrimental, especially if I suspected the parent might be abusive or on substances but had no evidence to report (or had reported to CPS, because we often get little/no response to that if it's minor).
For a book recommendation, I read and got a lot out of the book Teach Like a Champion. Silly title, but it's basically a group of classroom management strategies that have proven effective in urban classrooms. Some things in there just aren't me, but I definitely use quite a few things from it.
