I don't want to discourage you, but these seemingly simple questions cannot be practically answered in a forum such as this. On the surface, I'm sure the questions seem very simple. But the infrastructure required for a practical solution is quite complex involving network technologies (cloud gaming platforms?) and many other things.
Lua itself is a simple tool for scripting "structured programs" and "extending" and "embedding" the 'C' programming language. It comes with some libraries to do basic I/O and basic network functions and some other simple things. And the package manage 'Lua Rocks' has many other support libraries. But Lua itself knows nothing of games (or any other software domain) on its' own. All these things must be added by skilled programmers with the aid of other libraries - many of which have to be custom developed in 'C' or 'C++', or perhaps with a little effort in some other languages.
Lua is used to EXTEND many games because it is very easy to interface with the 'C' programming language. When way say it is used to EXTEND games, we mean that for a gaming application, it relies on a (possibly massive) underlying gaming platform that does understand gaming physics, graphics, and a myriad of other things. Lua is then used to "customize" how the gaming platform reacts to past and current player interactions.
Corona is also not a gaming platform, it is a cross-platform mobile device development environment. That is to say that before you can do anything constructive with Corona for a multi-player Internet game, you need to start building your server solution. Lua may fit into the server solution as well - but only in conjunction with a network API, probably a database of some sort, and a few other things. And you will no doubt need some additional libraries with Corona display graphics such as avatars, etc. And if you are envisioning this as an action game - the complexity is raised exponentially.
I don't know how robust you envision this game to be, especially with regard to the UX and graphics, but keep in mind that commercial games such as 'Sim City' and 'Call of Duty' are built by teams (sometimes very large teams) of programmers with many different skills. It sounds like you want to build a game combining elements of these types of games and may others. You will need to start with a good gaming platform to have any hope of success.
So you really need to decide on TWO gaming platforms: one for Corona and one for the cloud. Hopefully you can find a single vendor that offers both and are designed to work together.
To try to give you an actionable answer, I would suggest you start by doing an Internet search for "cloud based gaming platforms" and get an understanding of what is involved. And check out the "References" section here: https://en.wikipedia.org/wiki/Cloud_gaming
Then look for a vendor with a solution that can support what you want to do. Here is one you might look at: https://azure.microsoft.com/en-us/solutions/gaming/
There are many other ones, this was simply one of the first that came up based on my search terms. Once you settle on a cloud gaming platform, they may have lists such as this one, or other support mechanisms that will be more able to answer your questions.
As I said, I don't mean to discourage you, but am simply trying to provide a polite answer to help you understand that the questions you asked are simply too broad to be answered. As a beginner, you need to get a good understanding of structured programming and all the technology solutions available for your problem, and then you will be in a better position to decide if what you want to do is practical given whatever resources you may have available.