There are lots of folk better qualified than I to help you but I'll add my tuppence worth anyway.
As far as I can ascertain, a lamb is a lamb until 31st December in the year it was born - it then becomes a hogget - basically a big, heavy lamb How many terms for sheep does there need to be? Lambs in the UK are generally 5-6 months old at slaughter, so 20 weeks might be about right.
The time to kill a lamb depends on a combination of weight and fat cover - but fat cover is more important; it's better to kill a light lamb with the right fat cover than wait for it to get heavier, since much of the additional weight will be fat, which is waste. In the UK, commercial lamb carcasses are usually about 16-20kg, which is about half the liveweight. The killing out % is between 50% for a lamb and 40% for a hogget. A good carcass will have some fat but not too much and the way to determine the fat cover is to feel at for the vertebrae at the tailhead - the top of the spine will be well rounded but you will still be able to feel the bones with pressure; the side bones of the spine will be well covered and you'll have to press quite hard to feel them.
So, when to slaughter will depend on the breed, sex and feeding of the lamb - single lambs will usually be ready before twins or triplets; the meat breeds like Suffolks will be ready before the primitive breeds like Shetlands. Some folk let the primitive breed run on to 12 -18 months before slaughter to get a decent sized carcass (these breeds don't lay down backfat in the same was as modern breeds do).
We were given a formula for calculating the weight of a pig, using a bit of string - I wonder if there is a similar method for sheep, if you don't have a weigh crate...
Hope this helps - and I hope I remember it all for next year.