The replacements should be without fault. Both my Xperia's were replaced within the first 2 months of the handsets launch so they were both brand new.
Whilst I've given back a rooted Xperia, it wouldn't turn on so they couldn't tell. The replacement I received was given back the day after I'd got it, and I knew as soon as I turned it on it was going back so I didn't bother rooting etc. The waterproofing failed on the first and it broke after I was taking some pictures in my fish tank. Its replacement had a hideous yellow hue to the screen so I knew as soon as the Sony logo flashed on the screen, it wasn't a keeper and it was replaced the next day.
With the One X, the rotation sensor stopped working. I was rooted and running a custom ROM at the time, so I used an RUU (a simplified version of flashtool and ftf files that return HTC's phones back to stock) to flash back to stock to see if the ROM was the issue and it turns out it wasn't. With HTC devices, even after returning to stock, when you access the bootloader, it displays ***RELOCKED*** in big red letters at the top of the screen so it's blatantly obvious that the device has been modified. I went in to the Orange store, explained the situation, showed the guy the bootloader screen, I got a knowing smirk and he called their tech support for me. I explained the situation again, that I was rooted etc, they said it was a hardware issue and it didn't matter, they'd send someone round tomorrow. I chose a delivery window the following day and the courier was round with a replacement handset pretty much bang on when he said he would be. At no point during the process was I without a phone (the same for all 3 replacements).
That was way longer than I expected it to be, sorry
.
Accidental damage is fine, so technically, you could launch it off a wall and you'd be ok for a replacement. Re. the cover, I'd just be honest and say it happened before and that you didn't think it was worth the excess.
I didn't manage to get the exact case I wanted for mine, but I'm using this one at the moment:
Nillkin Super Shield Shell Case Back Cover for Sony: Amazon.co.uk: Electronics (Mine is black)
It has a small lip around the screen which offers a little protection from falling face down. I can confirm this works as I dropped mine on the tiled, kitchen floor at work last night. It's easy enough to take off if you have the charging dock, but I picked up a magnetic charging cable too so I don't need to mess about opening flimsy ports or taking the case off all the time etc.
Anyhoo, tl;dr, just give EE a call, be honest with them and I'm sure they'll be fine. If you are paranoid, just flash an ftf file to take you back to stock before hand.
PS - Vodafone have screwed me over in the past too, that's why I switched to Orange.