Sink drain flooding the toilet

We bought a new (1959) house in December and decided to redo the bathroom that had a pink toilet, sink, bathtub, and wall tiles. I got the bathtub in without too many problems. I've got the toilet in too,
but I haven't put the silicone around the base yet.
I finally got the sink hooked up today. The problem is
that when I ran the water in the sink, it made water bubble up from
around the base of the toilet all over the floor.
I don't really know what that means... My first thought is that there
is something wrong with the drain lines- a clog or something. But
before I had the sink in, the toilet worked just fine.
I'm guessing I'll need another new wax seal regardless, so I'm going to have to remove the toilet one way or another, but
after that, I don't really know what to do.
I've read a lot of posts that talked about vent lines clogging, and I think there is one on the sink line, but I'm not really sure how to access it to check for clogs.
If it matters, the house is on slab and has a septic system.
I'm really at a loss right now. Any thoughts?

So the toilet flushes ok?

You have two problems. The toilet seal is bad. This may be because you needed a thicker wax ring (i.e., the flange is not on top of the finished floor where it is designed to be - thus, too low in the floor), and you have a blockage somewhere. That is the only way the water would backup and drain out of the toilet seal. The toilet may not leak because it basically falls down the drain hole. I'm surprised, since it sounds like you have a blockage that flushing the toilet didn't do it, too. It may be that if you tried flushing the toilet when it was leaking, it would have been worse. That is because the water does drain, but slowly. Something you did previously filled it up so when you ran the sink, there was no more room in the drain.

If the toilet rocks at all, you will break the wax seal - the wax is not rubber, you squish it down once...if it rocks, you end up opening up a crack in the seal.