ok, lots of questions there, and you do need to sort out for future
WordPress consists of
core wordpress software
Themes
plugins
uploads
database
When you update “wordpress” – eg to 3.7, then it is only the core software that updates. Themes and plugins remain the same. this is why themes and plugins occasionally break, and they are ‘incorrectly’ using core software, and when that changes they no longer work.
Auto updating eg to 3.7.1 etc. is only for minor security and bug fixes, so should not break themes and plugins. Whether it ever will is open for debate. You can turn this off, but this is not recommended by wordpress. Auto updates are released in stages to ensure that they are stable. Major releases will still be under your control. So auto-updates should not break your site.
You method of doing manual updates (switching to another theme) actually does nothing to protect you – as state above the theme software is not directly affected.
Now onto themes.
2010 has had many updates – you should have been prompted recently to update it. If you have been doing that, then the 2010 theme is up to date. If you look in the themes bit of the dashboard it should tell you if the theme is up to date. These updates do not provide additional security (this is in core software) but enhance the theme or ensure it works with later core software versions.
You think you are on a ‘custom theme’ ie one that is not a deafult theme – even though it is actually an altered copy of a default theme – if you look in dashboard>appearance>themes it will say what theme you are using.
It sounds like you are, as if you’ve been updating twentyten, you’d have lost the changes the original fellow did.
If you copied the whole of the altered twentyten theme to a new directory, then your code will be safe.
However it is possible that future wordpress releases will break your theme. Until you fix this with a child theme, you will be at some risk, but it may be years or never ! Because Twentyten was written by wordpress it is a very solid theme, so you may be lucky.
So
No wordpress core updates won’t overwrite twentyten
You should check that you are using a custom theme – ie your altered twentyten theme.
You should keep the default themes up to date.
If a core update breaks your theme, the default twentyten theme will be the closet, and you will need to switch to that to get you site going again. then you will either have to live with that, or work out what the changes are.
If/when you site breaks, you will wish that you’d spent some time now working out what was changed. If you can it would be worth asking the fellow who changed it if he knows which files he changed. Chances are that he has not changed many files, most of the key look and feel are in a few places. He may have altered “header.php” that’s the one most people like to play with, and probably altered style.css, and maybe “page.php”. He’ll probably be able to tell you which even if he can’t remember what he changed. And yes, you ought to start looking to create a proper child theme, and then only having these altered files in it. you’d then be able to check if the latest twentyten versions have changed anything and keep these files up top date as well. This is NOT a lecture – as I know that it will take some time and learning to do this – I was there two years ago – and I expect your time is very limited.