John - When corks dry out, they become stiff and can be very difficult to insert into bottles with a hand corker. You don't want to soak them - that's generally a bad idea, especially if they are the common agglomerated corks (where the center section or the whole cork is composed of cork chunks, held together with some sort of glue). But it is a good idea to keep bags of corks well sealed before use, to keep them from drying out. If you got a bag of corks with a starter kit, they may have been handled by the retailer in a way that allowed them to dry out - the best thing to do at that point is to re-humidify them by storing them (dry) for several weeks in a high-humidity environment. Re-using bottles should have no impact on the corking process; if they are clean and dry, they should behave the same as new bottles, as far as corking is concerned.
Many people that get into this hobby end up buying a floor corker. They cost a little more, but just about anyone who has bought one will tell you it's the best investment they ever made. If it doesn't make sense to you to buy one, try to find one you can borrow for a day, when you need to bottle a batch.
Corks do come in different sizes; the most commonly used are # 9, either 1.5" or 1.75". This size seals well, and is generally easy to use with a floor corker. You can also find # 8 corks (both lengths) that do a pretty good job in standard wine bottles, and are easier to insert with a hand corker. If you're not planning to keep the wine longer than a couple of years, # 8 corks should be just fine.
The quality of the seal is affected by the length of cork in the neck of the bottle (among other things), so if half the cork is "hanging out", you probably don't have a very good seal (besides making your bottle of wine look pretty darned silly). If you have bottles with more than, say, 1/4" of cork "hanging out" of the bottle, you probably should remove those corks and re-cork the bottles. Or use them first, if the wine is ready to drink. :-)
Doug