Tips for building a soundproof room with ventilation
"Insulation" (in the ordinary construction sense) is essentially worthless for blocking sound. But if you need it for the conventional thermal reasons, then certainly go for it.

When it comes down to the bottom line, nothing BLOCKS sound except MASS. That means either the mass of objects (like walls and doors) or the mass of air (i.e. DISTANCE). 

And any gaps in the "envelope" will be HUGE holes in your noise-blocking scheme. So movable walls seem antithetical to the goal of blocking noise. 

Yes, mass-loaded vinyl is good for adding mass to block sound. But it is rather an expensive compromise in places where space is at a premium. A layer of sheet-rock dry-wall would have the same effect at a lower cost if you are on a budget.

It is not clear whether you already have a concrete floor?  If so, then that is a pretty good advantage.  Yes likely a carpet or rug to mitigate internal reflections. And that foam on the walls is not just "decorative".  Especially in a SQUARE room, it is necessary for reducing internal reflections. 

Note that making a space with equal size walls (or ceiling/floor) is a pretty BAD thing for recording sound because you are starting to create an "organ pipe" which will have resonance problems. Good studios not only have un-equal size walls, but they also have non-parallel walls so that "slap-echo" is reduced between parallel surfaces.

Note also that your choice of microphone will affect the effectiveness of your scheme.  If you use a modern "headset mic" where the tiny "wand" is right next to your mouth, you don't have to produce as sound to get a good recording (so not as much to block). And it will also have the added benefit of increasing the "signal-to-noise ratio) since your voice will be MUCH louder than ambient sounds (because of proximity), you won't have as much problem with ambient noise (including HVAC).

Not clear about your requirement for "ventilation".  Do you mean getting fresh air into the space? Or do you mean heating/cooling the air inside for comfort? There are different methods of reducing noise while doing both of those things, but few of them are very conducive for low-budget and/or temporary implementation.