The AX derailleurs had an early from on indexing which Shimano called NPM (New Postive Mechanism). The indexing was built into the derailleur itself, as opposed the shifters per SIS and STI. Your rear derailleur may work with 7 speed STI, but only if the cable travel per shift is the is the same as the levers. It is worth a try, but don't be surprised if they are incompatible.

The chain is not an issue if you stay with 7 speeds, even if you convert to STI, but it will be if you convert to more cogs. You may be able to get by with the current chain on 8 cogs, but it will be an issue with 9 or 10. The narrower spacing of the cogs requires a narrower chain. The current chain would be noisy or tend to skate on the cogs.

Finding actual AX lever hoods may be hard and/or expensive. I'm assuming you're in England and I really don't know what is avialable over there and the pricing, but your LBS should be able to set you up with something compatible. In a pinch I've gotten creative with cork handlebar tape. It's not the best looking, but it's quite comfortable.

There's nothing wrong with liking new technology. I've got bicycles with 4 cog, friction shifting derailleur systems, all the way on to 10 cog, indexed, ErgoPower (Campagnolo's version of STI). While I like them all, there are definitely some advantages to a combined brake/shift lever. One of the big supplemental advantages, should you decide on a new rear wheel, is the ability to have the ramped, Hyperglide cogs, which shift even under heavy load. You can be honking up a hill at low RPM and just have to flick your right hand to get easier gear. But don't make your decisions based on the opinons of myself or others. Borrow a bicycle with STI, try it out and decide for yourself if it worth the expense.