Hi - it's all very confusing isn't it!
So basically yes, we paid stamp duty and other final costs such as the commission to the estate agent, remainder of the solicitors fees out of the equity. The solicitor worked out the amounts and paid it all on our behalf and then there was a little bit left over that was paid back to us.
Regarding the buyer's deposit...so they paid a 10% deposit on exchange to us (well to our solicitor as they handle it), we then had to pay a 10% deposit to the person we were buying from (in our case, the builder). So the 10% our buyer had paid us, went to the builder as our deposit.
Also worth noting, is that 10% of the house we were selling, was less than 10% of the house we were buying. Our solicitor agreed with the builder that we could just send them the amount we got from our buyer, rather than the full 10% deposit. Apparently this is pretty common but worth mentioning to your solicitor in advance.
We were exactly like you, in that we had a fair bit of equity, but not a huge amount of savings therefore everything that could be done out of the proceeds of the house, was done that way.
I would estimate £300 for solicitor, £500 for survey, £200 for removal van. So if you have £1k, it should all be fine!
If you think of anything else, just ask as it's all quite fresh in my mind!