Can I make my own solo server? - Yes. Same exact concept as making a server for online use, you just don't open it to the public.
Can I update it when I want? - No. Each core supports a different version, so if you want to upgrade to BC from vanilla, you'll need to use a different core. The most popular is v1.12.1/1.12.2 (5875/6005), v2.4.3 (8606), v3.3.5a (12340), v4.3.4 (15595) and v5.4.8 (18414). Those cover vanilla through Pandaria. There really aren't any Warlords of Draenor cores that are any good yet, and Legion is much too new to expect anything good. The main core development groups are MaNGOS, C-MaNGOS and TrinityCore.
To elaborate further, fortunately the main core development groups (MaNGOS and C-MaNGOS) have a core for vanilla, BC and Wrath (whereas TrinityCore only does WotlK), so the setup is exactly the same between the cores and the databases are somewhat compatible (If you want to have your account from your vanilla server also on your BC server, you'll need to run the DB through a utility that will update it so it is compatible with the structure of the BC database).
As far as making the server goes, you have 3 choices. You can either compile your own fresh and up-to-date core, opt to use precompiled binaries from someone else or use a repack (A precompiled core with usually a lot of custom stuff, like items, creatures, NPCs, etc.). I'd suggest going with compiling your own core, it's easy to learn how to do and your don't really need much, if any, experience with compilers or SQL databases. The benefit is a nice clean fully up-to-date server.
If you want to learn how to compile your own core, check out our WoW MaNGOS Tutorials forum (
http://forum.ragezone.com/f518/). *Shameless plug* if you want an easy to follow tutorial, go with mine:
http://forum.ragezone.com/f518/mangos-server-easily-windows-719160/
If however you want precompiled binaries or a repack. Check out the WoW MaNGOS Releases forum (
http://forum.ragezone.com/f436/).
If there's anything else I can help you with, ask away!