I've been running a NC server for a couple of years now.

1. This is more complicated than it sounds - if you leave the server in sleep mode it may be possible to wake it with a "magic packet" ping, which comes with a new layer of complexity if you want it to be automated. I gave up in the end and just leave mine running - the little power draw to keep the little PC on seemed worth the trouble saved.
2. You can access it from anywhere if you have the server open to the internet. ( ie port forwarding through your router) . This gives you just a raw IP address on the open-side ( http://123.456.12.34 for example) which has all sorts of problems from technical to security: many services won't work with just an IP address, and without https protocol your connection is vulnerable to malicious activity. I started with just a raw IP address, but after some time shelled out for a domain name (10EUR/yr). This allowed me to access all services like music streaming, and to install security certs for proper https.

It sounds like a lot, but you can incrementally upgrade your system step by step and learn as you go.