These are my thoughts on choosing track day tyres for a Caterham. I have sets of 14" and 15" wheels. I don't know anything about 13" wheels with your low gears and ground clearance, or your tall diffs, or your tall wobbly sidewalls. I also drive my car to track days, so I'm only going to be talking about road-legal tyres.
Toyo R888-R
These are my favourite tyres, as long as there's not a lot of water. They're available in 14" and 15". They're about in the middle of the price range for semi-slicks. They have a very sturdy construction and stiff sidewalls. The grip they provide is excellent and you get quite a lot of use out of them. They're asymmetric so you can't flip them around on the wheel, this does mean that once you have worn out the outside shoulder the whole tyre is used up. The technical documentation for them says that the front suspension should have as much caster as the chassis will support. I think this is to try to get more of the width of the tyre on the ground during cornering. I have done this, but I still always wear them out on the outside.
Nankang NS2-R
These are a cheaper track day tyre. They're available in 14" and 15" They're very sturdy and have stiff sidewalls and you get a lot of wear out of them. They're directional so you can flip them over to even out the wear. I have only got on well with them when the weather is hot, when the circuit is bone dry, and when you've already warmed them up. In the cold they're terrible, and in the wet they're dangerous. What's odd is that people race MX5's on these in all weathers and they get on fine. My guess is that they will come up to temperature OK on a slightly heavier car, but that it's hard to get enough heat into them on a Caterham.
Avon ZZR
I've never used these. I believe that they're the top of the range semi-slick for our cars. I think they're 15" only. They're also too expensive, too hard to come by (I think you have to buy them from BMTR or Caterham), and I'm given to understand that they degrade thermally very quickly.
Avon ZZS
I've never used these. I believe that they're the top of the range all rounder for wet and dry. I think they're 15" only. They're also expensive and they're soft, so you don't get a lot of use out of them.
Uniroyal Roadsport 5
This is what I use in the rain. Only available in 15". The grip is pretty good, and they're pretty forgiving to drive on. They're not a particularly high performance tyre and they have very soft sidewalls. The whole car does feel rather detached and wobbly but you can corner and brake even in standing water.
Avon CR28
Available in 14" and 15". I've never used these because I'd wear out a set just getting to the track, but I believe they're the absolute best for wet conditions. They're a road-legal, soft compound, full-wet motorsport tyre.
Richard "Michelin Man" B