There isn't a site that saves all TikToks but there are a lot of forums that will have gotten it if it's someone that a lot of people follow.
No, you would have to download them yourself. There is no automatic downloading system when someone posts or records a video. You can download them many different ways. There are multiple softwares online to do so.
Do they play? After testing it it seems that they don't host the videos but simply use the Tiktok source where they function as a third party.
Deleted Tiktoks don't seem to load for me.
tik.fail was pretty great for a while. More of a repository of tiktoks that people would upload before they were deleted but it doesn't seem to update much anymore. Might luck out if you're looking for some older stuff.