1. Puting a search box should NOT affect at all to your rankings. Anyhow, that box will make some words, such as 'google' or 'search', appear on your page, and that might indirectly affect rankings for keyphrases containing these words. Most probably, these words won't interest you at all, but through the same idea; you can add links for some predefined searches to make some more interesting keywords appear in the file. In any case, if you try this technique, you should care about two things: 1st, you may be giving, through the links, PR to a competitor and 2nd, google might find that your links are artificiall and even ban you if you abuse the keywords.
2. This is very complicated: having a keyword on the domain (or subdomain) name may improve ranking for such a keyword, but it will harm you with 'site:' searches, 'cause they get limited to a domain name. IMHO, you should not center the decission about using subdomains or subdirs in SE factors.
3. This is the easiest one: the answer is YES. Google (specially the PageRank technology) cares about pages and not sites. If you make a precise search, you should not reach any homepage. Going further, one of the keys of Search engine success is to value each page and to know wich files should be enhanced for each keyword/keyphrase.
Hoping be useful,