Obviously you'll need a 10 speed chain and cassette, since 10 speed STI levers are not compatible with 7/8/9 speed systems, but the old rear derailleur will work with any number of cogs, adjusted correctly of course. The front derailleur will work ok, but it may be slow shifting into the big ring as 7/8/9 speed front der's have wider cages than 10 speed front der's to accommodate the respectively wider chains. Like I said, it will work, but it will take some careful adjustment and may still be a little slow shifting.
I'm guessing that downtube shifters with DA means you're probably talking about 7400 series stuff. If that's the case, your RD has a different cable pull to travel ratio than any other Shimano group (contemporary or current). 10 speed STI (regardless of whether 105, Ultegra, or DA) paired with a 740x RD will shift 9 speeds, with the last click locked out with a limit screw.
This setup actually works fairly well - after my old 6500 series Ultegra shifters died, I ran 7800 DA shifters with a 7402 DA RD across my 6500 9 speed cassette for the rest of the season before upgrading to a 7800 RD and cassette.
Depends on the DA rear derailluer. If 7400, DA 8s, they will not as this rear derailleur has a unique dimension and cable pull. If 9s, 7700 variety it will work fine. FD will work fine regardless. As another poster said, need a 10s cogset and chain...NOT crank.