There are loads of places on the net that will teach you the basics of what you're trying to do, but I'll toss in my $0.02 for you:
1. Yes, archive your collection. I can't speak to the vinyl side, but I've ripped my entire CD collection to an external hard drive that is connected to a Mac Mini via Firewire. This disc is easy to clone to a similar disc that lives in my fireproof safe. (Always back up!)
My CDs are stored in boxes in a closet, so that I don't have to worry about #8s ethical considerations. Yes, if you rip and then sell your CDs, you'd be violating copyright.
2. Dedicated server/streamers tend to box you into a proprietary solution that is not always better than a PC or Mac platform. In fact, with my external hard drive, I could swap various platforms and giblets in or out, if needed.
My choice is a Mac Mini running headless, addressed through any other Mac in the house. The Mini was modified by Mojo Audio to remove the original power supply, replacing it with a quieter, external supply.
3. For vinyl, probably. For CD, no, just a CD drive and software to rip. I use XLD to rip CDs directly into iTunes.
5. I'm surprised no one has yet mentioned this: I stream from Tidal. I find it to have far better quality than Pandora, et. al.
6. If you stream, you only need your computer and an internet connection (well, also DAC, speakers, amp, etc.) That computer should be dedicated to playing music, as you can easily disable all other functions that might affect the signal stream.
7. Costs for archiving are minimal. Costs for streaming are based on how much the services charge, like Tidal at about @20/month. (Totally worth it, in my opinion.)
Another thing you might want to consider is Roon, a relative newcomer to the market. Roon integrates your CD collection and streaming services such as Tidal.
My system is the Mac Mini and disc, iTunes + PureMusic outputting USB to my DAC, DAC to amp, and out to Magnepan 3.7. (There are, of course, many other bits in the system.) As mentioned, I have my collection of CDs ripped to a disc, and I have a subscription to Tidal.
Before Roon, I'd have to quit iTunes/PureMusic in order to stream from Tidal, and vice-versa. Roon, with its intuitive interface, non-destructively accesses your archived CD collection (iTunes folders) AND interfaces seamlessly with Tidal. So much so, in fact, that Roon will take a look at your collection and present albums and artists from Tidal that would likely interest you. The ability to play a given artist from your collection and to have Roon/Tidal suggest an artist that you may not have heard before... is a great feature. You'll lose yourself in making connections with your music.
Roon adds another subscription charge, but so far as I've seen... totally worth it.
You should also know that services such as Tidal feed the artists.