Iphone 4 and Mac OS X version 10.5.8

Question

I recently got this iPhone 4 version 7.1.2.  And it requires iTunes 11.1 on my laptop to connect with. But my computer is running OS X(version 10.5.8) and iTunes 10 is as latest as it can get. So they're not compatible at this moment.
so this is what I'm doing;
I email each song n book to my email acc. And I open them individually on my phone, then I m able to listen n read them through my phone. But they r still in my email acc, I am only able to see them through it.  I tried to send them somewhere using any apps on my phone, so that I can put them in one place together and I can get my listening n reading done at the same time. But I haven't found a way yet. So here are my questions.
1. Other than iTunes, is there any apps I can use? So I can download things thru my phone n store them in one place.
2. how do I back up? since my phone and computer, they are not working together.
Thanks in advance, I've been using Mac for quite a while (10 years), but I've never used anything like ipod, iphone, or ipad. so I'm feeling kind of new here.

Answer 1

Thanks Ralph, not quite sure bout the model of my laptop but this is what I found on hardware overview ;
  Model Name: MacBook
  Model Identifier: MacBook5,2
  Processor Name: Intel Core 2 Duo
  Processor Speed: 2 GHz
  Number Of Processors: 1
  Total Number Of Cores: 2
do you think this prob will be solved if I raise my Mac to Snow Leopard?
and I have more questions ;
1. I don't think I have iTunes on my phone, it's weird. I used "spotlight search" and typed "iTunes" and nothing came up. 
2. other than iTunes, is there any apps I can use? and it doesn't cost or charge anything when downloading and installing.  It's gotta be something, but I haven't found yet.
Thanks again Ralph!