safari won't identify itself properly, xml parsing problems and more

Running Safari 5.0.6, OSX 10.5.8 (don't ask - it's the latest I can use), Intel iMac.

Sometime this fall Safari just stopped working properly - I've tried reinstalling, without luck.

Here's what happens:

1) Many websites tell me that I'm using an outdated browser or that I'm using a browser they don't support, then instruct me to obtain a copy of the browser I'm using. They won't allow me to access their site. I've tried enabling the develop menu and forcing Safari to identify itself as something, but it doesn't help.

2) Often I'm told that my browser isn't accepting cookies - when it is set to do so.

3) Many websites just don't work right. They work fine for me with Chrome, or Opera and usually Firefox.

4) Some web sites just crash with "this page contains the following errors..." which I take to be xml parsing errors. This problem (on the same sites) also happens with Firefox and Chrome (and sometimes Opera, though Opera will work since it just re-parses it as html and it's usually OK). It never used to happen, and indeed I can go the the site with an old machine (OS 10.4.11, and Safari 4.x) and it works just fine. Perhaps there are indeed errors, and Safari is just picky where it didn't used to be - but then there's no way to recover - and these are large, well-visited websites, not something rarely used, so if there were serious errors, they'd be noticed and dealt with sometime in the last 6 months).

I don't know whether this is related to the above problems or not. Since it's not limited to just Safari, I have to assume it's in some sort of shared parsing code.

Why don't I just not use Safari? Well, there's some things I actually like about it, and my wife prefers it to the others because she's used to it, - and then there's the parsing problems.

So - any ideas?

How can I identify and reinstall any shared libraries these browsers use?

Thanks for any help.