candles, all lights completely off, wine for pleasant or sombre or soft vocals, tea for earthy, or crisp vocals, or whiskey for hard edged vocals.
A relaxed environment, with an open minded attitude as to however it's to be done, so as to make the vocalist comfortable. An understanding of the message/story of the song, and how one might coach (in a relaxed way) a vocalist to best execute that. It's kind of like juggling, and it's a skill that you may or may not pick up, but will only improve your vocalists' takes. Voice is the one instrument that can't lie, and if people don't believe what the vocalist is singing, it's ineffective.
note: wine is more for setting the mood than having any effect on the vocals. Whiskey is nice if you're really into doing some hard edged shit where your vocal cords could used to be fucked a bit, and of course we all know what tea is like.
I tell you, you do the more psychological techniques, and no matter what mic you use or where you put it, you'll notice an improvement. I'd rather have a great take than a great mic any day. Find the sweet spot with any singer where they tend to give their best takes, and make sure to get into that zone as much as possible. Everyone has their own zone. Some people do the best on the first few takes, some people it takes a while to get going. Some people like it rapid fire cycle recording, some people more prefer to have it slower paced. Some people the entire song, some people sections at a time. Just use your intuition (that's more of the name of the game than anything someone could teach you about recording vocals), and make sure, with everything you do, to keep the psychological concerns at the front of your mind. If you're having technical problems, try to play it as smooth as you can, not be visibly stressed or panicked... just say, okay I'm just doing some last minute technical stuff before we get going...would you like a drink? Never rush a vocalist, or act like you are pressed for time. Just create a really good atmosphere, that is one that someone could be very comfortible to explore the full capabilities of their voice, without inhibitions, and this also creates an atmosphere of openness, that will make it easier for you to coach the vocalist, without them getting defensive (trust me...you want to coach the vocalist... you really want to... You should know how it aught to sound, so, without being to invasive...coach the damn vocalist.)
Other than that, put a mic that's not too shitty in front of the vocalist with a windscreen/pop filter, using your ears to pick a spot that doesn't sound TOO terrible, and hit record. If the vocalist is the right one for the recording, you'll get good results, even without thinking of mic choices, preamps, cables, etc.. (those can always help, but it's MUCH more important to achieve the other stuff first, before you even think about the technical end)
As far as the specific questions. Put the mic in another room and close the door... get long cables...and a long headphone cable... line of sight is nice, but it's better to be rid of huge amounts of noise. Set up a mic in your room that goes to the headphones so you can talk back to the other person... or you could just walk into the other room between takes (that's what I do, I just think it makes it a bit more personal to work with the vocalist face to face and give them some bit of physical feedback between each take..that's just the way I work tho.. you could do a talkback mic)
The hum could mean...a bad mic, bad cable, or grounding issues. Make sure your stuff is properly grounded. My alcohol ridden brain can't quite remember all of the particulars from those electronics classes I took, but just do a web search and it should tell you how to get rid of hum (get rid of it before you record, not after)
A closet can be a bit too small of a space, in my opinion. I've never had great results with them, or with bathrooms. If it's too cramped, it might sound more boxy, and "lumpy".
Get a quieter computer/or try some liquid cooled solution (my computers are quiet enough that I could record in the same room, and it probably wouldn't show up on a recording unless I compressed the ever living fuck out of it, more than would sound good anyway (the inherent noise of the mic would be about as much of a problem in that case). My G4 would be way too loud to record anywhere near, so if it were in the same room...off it goes... or, I just record as far away from it as poss.
