This is one of those, "if you have to ask" scenarios. If the little voice in your head is saying maybe this isn't such a good idea, maybe it's not such a good idea.

Use flame retardent foam, not just styrofoam.

Id say you are fine, but I would add a fire extinguisher like others suggested, just in case. Many people have cheaper printers in ikea lack enclosure, which is made of cardboard, but they are not good example. Proper solution would be a metal enclosure and you being there while printer is on. But come on, who got time to watch printer for 20h or so.

If noise is too high in next room, you can reinforce your cabinet with some metal brackets or stop vibrations with rubber or simmilar. You need to find where noise is comming from. Also, you can add acrylic door (with holes on top/bottom for airflow), it should reduce noise significantly. Put thermometer inside and check temps after long prints. If enclosure gets too hot add more holes or a fan.

If you put insulation inside the cabinet, use some screws to make sure it doesnt fall off from the heat

I would also recommend an anti-vibration mat (typically used for washing machines).

Doubtful. The main reason I could see a 3D printer starting a fire is either thermal runaway, often caused by bad software (on cheap printers) or electrical issues. You can check your wiring and make sure that nothing looks loose and all of your cables are in good condition. You can also make sure that your printer firmware is up to date. It's a Prusa right? I think you should be safe from software failure in that case, as they're quality.

If you're really concerned about it, you can also move your power supply, and even your mainboard, outside of the enclosure by using longer cables. This would keep the majority of the electrical components away from the foam and heated box. From my position I don't see an issue, but that's my personal opinion -- not legally accountable information. I will say that Stefan on CNC Kitchen uses foam as well, to get rid of ringing in prints. If you decide to go through with it, just use common sense and check to see if anything is hot. You might also get some benefit out of looking into silent stepper motors/drivers and trying to make the printer itself quieter.

In addition to the stuff other people have pointed out, make sure you glue your insulation down very well, so that it won't fall off and come into contact with your print head. Just in case. Other than that, I don't see anything other than the usual potential electrical issues.

A lot of noise comes from transferring vibrations into what it's sitting on. You might consider isolating it may be more effective.

https://youtu.be/OnfYA5QLA84

I'm not an expert here, but looking at the volume of the space, you're probably going to be fine. Why don't you add a small fan in the back of the enclosure to get some more airflow?

Am I building a fire hazard?

Short answer: yes

Long answer: yyyyyyeeeeeeeessssssssss

It gets worse and worse :)

Yes, using a high-power electrical device inside your wardrobe is a very efficient fire hazard. It not only makes it easy to start a fire, but it also provides plenty of fuel to jumpstart over the iffy small-fire stage where it still has a chance to not consume your entire home.

Also, sending those smells somewhere they won't harm anybody is a significant concern when you decide where to place a 3d printer.

Just put an automatic fire extinguisher in there to be safe. These are common in apartments without sprinkler systems.

https://a.co/d/j0lRBOC

Do you close the door? If not, I'd think you'd be fine. Even if you do close the door, it might still be okay, idk.

I saw someone else also suggest it, you could do an automatic fire extinguisher.

This one dispenses after 3-5 seconds of direct flame contact, others will be different.

if you put a clear door on the cabinet you can suffocate a fire

Add propain for extra damage!!

LOL, it should be fine just make sure to check bed wires every once and a while. My printer did spooky things

It's a Prusa Mini, so probably not. Quality parts and thermal runaway protection and all that..