You can use the following command to tell you how much space is available on your VPS:
Your host can also over provision as well, so this will only tell you how much space is allocated to you, not how much diskspace the parent node has available.
You may also want to check the disk inodes with this command:
This will tell you how many files you can create (and is sometimes artificially low with some hosts).
Edit: To answer your updated question:
VZFS is not really a file system as you're thinking about it. It sits atop an existing file system which is most likely ext3 (the per container disk quota system will only work on ext3 and maybe reiserFS if I recall correctly).
If your host is running Virtuozzo (the commercial openVZ) it will help the host save diskspace by sharing files that are the same between containers (e.g. if all containers have the same file "/etc/foofile", it will share it amongst the containers. That's about the extent of the magic, if you look at the containers serverside you'll see regular files, there are no container 'partitions' or images preformatted and you're free to move files in and out of VEs with mv/cp.
In short - you're probably running on ext3 underneath. You can do what you want, although keep in mind most VPS providers I've seen are not geared for handling very large databases, you might be better off on a dedicated server unless your database is well optimized.