What is a demiplane?
Demiplanes are defined as "small extradimensional spaces with their own unique rules" (in the PHB's appendix C, p302, they're also described with slightly different wording in the DMG, p68). So any small plane can be called a demiplane, and you shouldn't expect them to all be the same or have even have many rules in common. The whole Ravenloft campaign setting is contained within the Demiplane of Dread, so even "small" is relative.
Creating demiplanes with spells
In your question, you seem to be specifically referring to demiplanes created by the 8th spell Demiplane (PHB p231, in basic rules, and online here). It's worth noting that that's not the only way to create such a plane. Several other spells also create demiplanes, some of which are much simpler. For example, the 2nd level spell Rope Trick (PHB p272, in the basic rules, and online here) creates a temporary demiplane at the top of the rope that is only defined well enough to say that it can fit up to eight medium-sized creatures. The 7th level spell Mordenkainen's Magnificent Mansion (PHB p261, also in the basic rules as Magnificient Mansion without the proper name, and online here) lets its caster conjure up a whole building with furnishings and invisible servants that exists in a temporary demiplane.
The demiplanes created by the Demiplane spell have some specified characteristics (30ft cube, bounded by wood or stone walls, and since you can come back to the same demiplane, it's presumably permanent), but other characteristics are unspecified (like how a created demiplane relates to the Ethereal Plane). There are no official rules yet in D&D 5e for joining demiplanes, or many of the other things you have asked about. However, many of those ideas ripe targets for homebrew between a PC and their DM.
Homebrewing special rules for demiplanes
If you're the player of the PC who wants to play around with demiplanes, I suggest taking to your DM about exactly what you want to achieve. Together you can work to come up with an appropriate set of rules. For ideas, consider looking at earlier editions of D&D and Pathfinder, where you'll find playtested and at least somewhat balanced spells and other rules (though as with any homebrew, your DM will need to think about how well outside concepts will fit into your specific game). This might be one good starting place!
Your DM might tell you that you can just do whatever you have come up with, as part of casting Demiplane. Or maybe you will eventually be able to learn to do it, but only after a long-term project, as you (or NPC wizards that you hire) will have to do a bunch of research and experimentation first. The learning process might be something you come back to several times between your other adventures.
But your ambition to tinker around the defined boundaries of your spells also gives the DM a great hook to send you off on an adventure specifically about this topic! Maybe your preliminary research will tell you that you can't merge two demiplanes without the knowledge of a specific wizard who once lived in your world. But alas, she's been lost in extraplanar space for decades or centuries. Can you find her? If she's trapped on a demiplane with weird rules of physics and time, can you get her free without becoming trapped yourself? Will she agree to teach you if you do rescue her? It could be fun to find out!