1. Safari's support?
Safari is actually a decent browser. If it has its flaws, they aren't any worse than those of any other browser, and they aren't of the class of the old IE browsers, which had very serious problems and lacked even basic support for web standards. To answer you question specifically, yes, it does support absolute positioning.
Safari can certainly render modern X/HTML CSS designs, and since your audience is largely using Safari anyway, you may as well forget the notion of dismissing the browser. It's a good browser, and in any case we're powerless to change it. We simply need to take care of these bugs, whatever they are.
2. How to go about debugging?
Without having a specific example, it's not something anyone can really help you to do. It seems fair to say that you're having some issues controlling css-based layouts. You may have some invalid markup, which in some cases could produce the kind of extreme browser-specific abnormalities you've described.
Start with the basics. Validate your markup and CSS.
Make sure you're rendering in standards mode.
Seek out answers to specific questions
If everything validates and you're still having problems, you'll have to track them down one by one. Even if you rebuild the page, piece by piece in Safari in order to see where things begin to unwind, it will be worth it to do. If during this process you really don't understand why a certain behavior exists, you'll at least have a specific question that you can use to poke around for answers. It may be answered already on SO, and if it isn't, you can ask it.