1) You'll want a datacentre if you desire to Co-locate one of your servers somewhere else for fast internet access.
2) You can order a colocation spot the same way you'd order anything else online. With your credit card and thier web site.
3) Somewhere between US$60 and US$200 per month would catch all datacenters I've ever seen.
As per Alberto, you could always get some server lines installed to your house, but a datacenter has redundant electricity, fire controls, 24/7 techs, redundant networks, et cetera. Generally a good idea to use a Datacentre.
A lot of compaines wont install any lines in Residental houses, also the very fews times they do they price gauge you. I had a quote for hard fiber to get laid it was like 75$ foot plus all these other crazy fees was going to end up being around 10k to get it laid and around 4k just for a T1. I kindly said no thanks, it was to our "Business" in house.