I think Nyororin can answer more about the non-teaching lifestyle and job opportunities better than I can, but I will give you my opinions/answers.
How many foreigners do you encounter on a daily basis? And of what nationality are they usually?
During the week I see absolutely zero gaijin because of my location. I am in a school that is on the outskirts of a rather large factory town (hometown of Subaru). I know a lot of English teachers that are spread out across the city and live north of me in a larger urban area. Somewhere like Tokyo, especially Roppongi, you will see a ton of gaijin everywhere you go (almost). How I met them all was that basically you get introduced through other teachers. English teachers, especially ALT's, are fairly close knit and tend to try to take care of each other. JET's, the ones hired through the government, are even more so. JET's have many parties and events that are held for them as opposed to the private ALT like myself where you're company drops you off at your apartment and shows you the school and says good luck. But all of us, JET or private, try to take care of the newbies the best we can by sharing info and socializing on the weekends. Sometimes though, I honestly would rather go it alone. A lot of the sightseeing I want to do many have already done so I just take off on my own.
Nationality of English teachers is varied but the majority are Brits, Aussies, Canucks (Canadians), Americans and New Zealanders. There is a small group from South Africa and then others that are from places like Denmark, Finland and other places like that. Those from non-English speaking countries face a little more trouble being hired as English teachers, but there are a few.
Also do you have alot of foreign friends? (and do they speak Japanese?)
Something like 90 percent of my friends are foreign just because I meet them easier, but even then we usually have Japanese friends who hang out with us too. Japanese level varies a lot but I think only a small percentage are fluent or close to fluent, majority of us have broken Japanese and there are those who know none at all.
Which do you think is the best city to start off in? Or live and work in once you're settled?
This depends a lot based on what you want to do. A lot of people say don't live in Tokyo but competition is pretty high for English teaching jobs there. In my case I chose to live in Gunma because it was as close as I could get to the ski resorts. I would love to live in the Kyoto area, maybe even closer to the mountains like Nagano or Niigata or more central Honshu, but I am fairly happy with my locale. If you go the JET route, you get placed in your school so you don't have a choice where you live.
I also read you can get free Japanese classes once you're there. I think that would help alot (even if I spoke abit of Japanese once I get there)
Many community centers offer these classes and if you work for a larger to mid-size private company they offer Japanese classes. The community centers are not free but are low enough to be almost free.
Concerning jobs. This might sound weird, but would someone turn me down for a job just because of me being a foreigner? (even if I speak/read/write fluent Japanese?) Have either of you ever experienced this?
You need to take the JLPT and acquire a level 1 for the most opportunities or level 2 for a fairly good chance. Here's the site
JEES Japanese Language Procifiency Test Home
Ganbattene!
One more thing about teaching. Hiring of new ALT takes place three to four months ahead of the start time. Starting time for most positions is in August/September and April. JET's have to apply and interview a bit early, I would say five to six months before start time. JET's get paid more (300,000) and don't take a private position that promises less than 250,000 yen a month. You will have a hard time budgeting for anything less than this.