Certificate to a Sub domain on my home server

Solution 1:

Is this possible?

Yes. Simply request a certificate from Let's Encrypt for dev.example.com. This can be verified using e.g. DNS challenge, or some other type. What matters is that you can respond to the challenge successfully, and you'll be handed a certificate.

If so, should the cert be installed on 2.2.2.2? Does the cert need to be setup in a certain way?

Probably. The alternative would be for all TLS traffic to be terminated on a reverse proxy at some other machine. But the most straight forward way is probably to terminate it at the machine that runs the actual server software.

There is nothing special to the certificate. There is no inherent difference between example.com and dev.example.com, or www.example.com; They are just domain names. Domain names are written in to the certificate.

IP's are not written into certificates. You can freely change where DNS points, and the certificates will continue to work.

Bonus question: Does the complexity of the setup in my home network matter? For instance: I have a router that directs port 80, 8080 and 443 to a VM web server. This server then use Proxy / Rev Proxy to redirect traffic - based on folders (eg. dev.example.com/app34) to the specific Application server (192.168.0.34). Will any of this affect the cert setup?

Whatever service that is expected to understand the traffic has to terminate the TLS session. This will in your case be the reverse proxy. Otherwise it can't decide where to forward traffic.