Web Server

If you're going to run your own server at home, there are a bunch of factors you'll need to consider. Do you have a static IP or dynamic? If the latter, you'll need to look into a dynamic DNS service as well. Is your internet service residential or a business class type of service? If it's residential, do they allow customers to run servers on their network?

Your internet service speed is also a consideration. You are anticipating a sizable volume of traffic, and most likely your internet upstream is considerably slower than your downstream; i.e. 50Mbit down and 5Mbit up. Your upstream is the traffic you are going to serve, meaning that when someone accesses your website, they are going to be limited to the speed of your upstream, and are competing with others for it.

If you are going to host a pretty active site, you're also going to want to be concerned about downtime. If your power goes out, how long can you last on batteries? If the power goes out on the whole street block or part of the neighborhood, will your internet service still be active? Your ISP may not have backup power on the local street hubs.

Besides the questions you are looking for answers on, of the type of computer and operating system, there are many other factors to consider - especially if you are planning on a higher volume site. For that reason, I would strongly recommend you look into colocation service or rent service space somewhere.

Well an IP address is a unique set of numbers for a device on the Internet, and having a static IP means it will never change. It's like having a mailing address for a house, or a phone number. It's the number identity on the Internet you can be reached at. Whatever domain or subdomain you use (like mydomain.com) will be linked up to an IP address.

The ideal setup for running a site would be to have a static IP. You can contact your ISP about getting a static IP if you don't already have one. If you don't have a static IP address, you're running on a dynamic IP, which will change from time to time as it's released back into the pool of IP addresses on that network. If you use a dynamic IP address, you'll need to use a third party service with software on your computer that will always ping the dynamic IP service and tell them your current IP. The dynamic DNS provider will then update the records of your domain with the current information.

You said your speed is 45/5. Again, I would recommend you put your server on a host somewhere with a faster connection if you are planning on pretty high volume of traffic. It is not the 45 Mbits users will be accessing, it will be the 5 Mbit upstream. Your upstream is their downstream, and filling up that upstream will also affect your 45 meg downstream, as you wait for requests to leave your upstream.

Being in NYC, you have quite a large backbone of providers running through there and quite a few companies that allow colocation (putting your server in their rack or closet), and many will allow you to physically access it when needed.
If you go with colocation though, you're going to want to find the smallest or slimmest server you can find. I would suggest looking into a Mac mini for instance, you can find them at a good price used even, and they can run Ubuntu linux as well.