You've been given a lot of good information already. Basically, your first question is divided up into two parts. If you want your own domain (yourcompany.com for example) then you have to register it. There are a number of companies that will take care of that for you. If you're connected to the internet you likely already have an Internet Service Provider (ISP) so check with them first. Registration varies depending on what country you're in. In Canada for example registrations are managed by CIRA. That gets you a name. Owning that domain name will cost you an annual fee.
The second part is finding someone to 'host' your domain for you. That means that the files live physically on their servers, and people access your site by connecting through them. Again, you may wish to check with your ISP to see if they offer domain hosting. This is where it gets tricky because costs vary with the amount of storage space you desire as well as your monthly 'throughput'. Basically throughput is the amount of information that is traveling back and forth through your site. To use a really simple example, let's say you have a 6MB file on your site (hosted on their server), and 100 people download this file in one month. That's 600MB of throughput. That's also why people use much smaller files!
From there you have to decide if you want e-mail addresses associated with this domain (hosted on their mail servers), how many e-mail addresses, throughput for e-mail, etc. Generall speaking, companies will offer packages offering x amount of hard drive space, x amount of throughput and x e-mail addresses for $X/ month or /year. This is in addition to the domain name registration. Some things to keep in in mind are what kind of connection they have to the net's backbone (T1, fibre optic, etc), what their connection speeds are, what their user/node ratio is (imagine 3 people waiting to use the bathroom at the same time and you discover why a high user/node ratio isn't a good thing), and other things like that.
Your website itself varies considerably depending on your own programming knowledge or if you want to hire someone to design it for you. A simple Lightroom gallery will get you started. Flash is a tool like any other. I think it has it's place and is in common enough usage (flash player is a free download) that you could consider it. As has been mentioned however, overuse of flash or other software can make your site look garish or gaudy and is likely to turn people off.