I believe everything you want to do is reasonable, but there's some details I'm hazy on.
You don't want to "run your own server": you want "shared web hosting". My vendor of choice for that these days is [REDACTED - will tell you in DM] and I highly recommend them. I'm paying them $39.48 per year for unlimited domains, and unlimited subdomains at those domains (and all the sub-sub-subdomains unto whatever is the actual protocol limit on such things.) Tremendous price, excellent features, excellent uptime, and superb tech support. They have never insulted my intelligence, even when I have been a doofus. Also, they have been super tolerant about the weird-ass things I do to/on their servers.
Like ~all shared web hosting these days, they run CPanel. This is an industry standard web-based interface to account configuration and application deployment. They also support ssh if you want to get in there and vi your config files for yourself.
Now, as it happens, because I do it myself, I know that it's possible to, instead of having "subdomain.mydomain.tld" point at a directory on the same box as mydomain.tld, have it resolve to a server on the other side of the planet. I have www.mydomain.org and mydomain.org with another hosting company (with other features) in Europe, and lists.mydomain.org along with mybusinessdomain.com, myprofessionalidentitydomain.com, and several other domains, over with this CPanel shared hosting vendor; lists.mydomain.org actually points at my share of their Mailman server.
As to how you would actually do this with DW or Medium, I don't know. I've seen this done for LJ, but I don't know if DW supports "embedding", as they used to call it. I should find out. I'm pretty sure it's possible to slap a domain on one's Medium presence – I think I've seen it done – and if it's possible to do a domain, a subdomain shouldn't be a problem, unless Medium has done something to implement the feature wrongly.
For buying domains, I recommend Namecheap. They are remarkably non-evil for a domain registrar, though their tech support is kind of mediocre. User interface for their service is quite nice.
I would recommend that if you have a domain in mind, you hop-skip to go buy it immediately before anybody else poaches it, and then go figure out how to deploy it at your leisure. You don't need to use a domain you've paid for, or a have a place to serve it. You can buy it and not do anything with it.
ETA: Things you care about in a domain registrar (that's the technical term for a service from which you buy a domain):
1) NOT NETWORK SOLUTIONS. 2) They won't go out of business abruptly leaving you and your domain in some weird legal limbo. 3) They don't have a reputation of either doing evil things to extort money out of you, or make it hard to migrate out of the service, or steal your domains from you or make it easy for others to steal your domains from you, or just spam you into oblivion. 4) Tech support by phone and/or real-time web-based chat.
You should be aware that when buying a domain, it's like a deed: it's a public record. Your name will go into the WHOIS database as the owner of the domain, and you must have a phone number, email address, and mailing address of record there. HOWEVER, there are privacy services, where for a small (~$2/yr) fee, a proxy company will hold your domain for you and put their info in the WHOIS database.
The pros: The WHOIS database is not just harvested by spammers, but fraudsters who send fake bills through the US mail. Having a privacy service largely defeats that.
The cons: 1) The entirety of your legal record of your owning your domain is your WHOIS listing. If your domain registrar goes belly up, and your name is not the one on the WHOIS listing, how will you prove it's yours? This actually apparently happened, and was a huge problem for the people who had used privacy services; maybe it's now a solved problem. 2) While this allows you to register a domain anonymously, if you ever want to move registrars, you will have to expose your real name/record to do so, apparently.
no subject
You don't want to "run your own server": you want "shared web hosting". My vendor of choice for that these days is [REDACTED - will tell you in DM] and I highly recommend them. I'm paying them $39.48 per year for unlimited domains, and unlimited subdomains at those domains (and all the sub-sub-subdomains unto whatever is the actual protocol limit on such things.) Tremendous price, excellent features, excellent uptime, and superb tech support. They have never insulted my intelligence, even when I have been a doofus. Also, they have been super tolerant about the weird-ass things I do to/on their servers.
Like ~all shared web hosting these days, they run CPanel. This is an industry standard web-based interface to account configuration and application deployment. They also support ssh if you want to get in there and vi your config files for yourself.
Now, as it happens, because I do it myself, I know that it's possible to, instead of having "subdomain.mydomain.tld" point at a directory on the same box as mydomain.tld, have it resolve to a server on the other side of the planet. I have www.mydomain.org and mydomain.org with another hosting company (with other features) in Europe, and lists.mydomain.org along with mybusinessdomain.com, myprofessionalidentitydomain.com, and several other domains, over with this CPanel shared hosting vendor; lists.mydomain.org actually points at my share of their Mailman server.
As to how you would actually do this with DW or Medium, I don't know. I've seen this done for LJ, but I don't know if DW supports "embedding", as they used to call it. I should find out. I'm pretty sure it's possible to slap a domain on one's Medium presence – I think I've seen it done – and if it's possible to do a domain, a subdomain shouldn't be a problem, unless Medium has done something to implement the feature wrongly.
For buying domains, I recommend Namecheap. They are remarkably non-evil for a domain registrar, though their tech support is kind of mediocre. User interface for their service is quite nice.
I would recommend that if you have a domain in mind, you hop-skip to go buy it immediately before anybody else poaches it, and then go figure out how to deploy it at your leisure. You don't need to use a domain you've paid for, or a have a place to serve it. You can buy it and not do anything with it.
ETA: Things you care about in a domain registrar (that's the technical term for a service from which you buy a domain):
1) NOT NETWORK SOLUTIONS.
2) They won't go out of business abruptly leaving you and your domain in some weird legal limbo.
3) They don't have a reputation of either doing evil things to extort money out of you, or make it hard to migrate out of the service, or steal your domains from you or make it easy for others to steal your domains from you, or just spam you into oblivion.
4) Tech support by phone and/or real-time web-based chat.
You should be aware that when buying a domain, it's like a deed: it's a public record. Your name will go into the WHOIS database as the owner of the domain, and you must have a phone number, email address, and mailing address of record there. HOWEVER, there are privacy services, where for a small (~$2/yr) fee, a proxy company will hold your domain for you and put their info in the WHOIS database.
The pros: The WHOIS database is not just harvested by spammers, but fraudsters who send fake bills through the US mail. Having a privacy service largely defeats that.
The cons: 1) The entirety of your legal record of your owning your domain is your WHOIS listing. If your domain registrar goes belly up, and your name is not the one on the WHOIS listing, how will you prove it's yours? This actually apparently happened, and was a huge problem for the people who had used privacy services; maybe it's now a solved problem. 2) While this allows you to register a domain anonymously, if you ever want to move registrars, you will have to expose your real name/record to do so, apparently.