Can I put videos on my iPhone from a different computer without erasing the pictures on my iPhone?

Yes, when using iTunes to sync, the exact content of your mac or pc is copied and missing files erased… Thats the way the files are managed on the iPhone…. Maybe check for third-party software?!

What about photos taken with the iphone? Will they be erased?

you can email or copy the file to your iTunes them update your iPhone

photos taken wont be erased, no :)

Set your itunes up so that you can add files manually instead of by syncing them. When you sync your phone, it replaces the old stuff with the new stuff on the computer you’re using. So click the box that says “add files manually” or something like that. That way, you can drag and drop files into the playlist from the folder you’re getting it from on your hard drive, and you won’t lose a thing. I’m not too sure about the pics tho. You might wanna back up the whole thing first.