See below -- I hope this helps. It's great that you are learning to cook! Just be warned, it gets addictive.

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1) 'Skillet' - what is this? Like a frying pan? I have a small kitchen, and don't really have room for any new cooking apparatus

Yes, it's basically the same thing as a frying pan, unless the recipe specifies a cast-iron skillet. I personally don't do cast-iron because you aren't supposed to wash it and that grosses me out, so if the recipe calls to bake something in the cast-iron skillet, I just use a baking pan instead (8x8 or 9x7 usually works). If it's supposed to be fried in the cast-iron skillet, I just use a regular non-stick skillet.

2) 'Stockpot' - I don't know what one is or have one, can I just use a large saucepan?

A stockpot is a large pot (usually several quarts at least) -- large enough that you can stick a chicken carcass and stew the heck out of it to get chicken stock. They are typically "flimsier" feeling than a regular pot. I have one but rarely use it, I typically just use one of my larger pots instead. The big thing is to make sure that the pot you use instead has enough volume to handle whatever was supposed to go in the stockpot.

3) Homemade applesauce - if I make a big batch in advance, for cooking over the next couple of weeks, how should I store it - airtight container / in the fridge / freezer?

Ooh, I'm not sure about this one. You do need to at least refrigerate it, but I'm not sure how long before it gets moldy. I have a feeling that fresh applesauce might get moldy in a hurry, but I'm not sure how well it would freeze. I'll have to defer to someone else on this one.

4) 'Baking soda' - what is this and can I get it in the UK? I have Bicarbonate of Soda and Baking Powder, but nowhere can I find baking soda.

Same thing as bicarbonate of soda.

5) 'Egg beaters', and other egg white substitutes - are they as nice as using proper egg whites? I'm going through so many eggs it seems such a waste throwing half of them away every time! My boyfriend seems to think they'll give nasty results...

I've never tried any of the dried egg subs, but I've had Eggbeaters and other similar liquid egg white products, and they usually work pretty much the same (except when you need to beat them stiff, like in meringues or homemade angel food -- they won't work then), although to me they seem more liquidy. They are much easier and less messy overall than eggs. However, over here, it is much cheaper to buy a carton of eggs and just throw out the egg yolks than it is to buy Eggbeaters. If you feel wasteful, you can freeze egg yolks -- just make sure to stir in a pinch of salt or sugar or they will get all yucky. Thaw them out in the fridge and use the yolks to make something hideously fattening for your non-IBS family/friends, like creme brulee. I used to freeze the egg yolks and use them when I was making cookies to take in to work or someplace where I wasn't going to eat them anyway.

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