Well it sounds like you will not be using a full drumkit in a room that size, so that is one blessing.

A false ceiling should not be square to the walls or floor. Reduces Standing waves.

Eggboxes reduce reflection - but they do not much affect the transmission of low frequencies unless filled with something pretty solid. Try expanding foam. THis is a guess.

It will be really hard singing in a room with no ring. You may want to add a frame to enable you to slap up a chipboard or plywood (different sounds) false wall on two facing walls to add ring.

In a room that size you will tend to get guitar (etc) on the vocal mic and vice versa. Work VERY close to the vocal mic and if that means you overload a C1000S (or better, a 3000S which are being remaindered as I write) try an ATM89HR (phantom power required, but cheaper). Or if you want to go dynamic try an AKG D880. Warmer than an SM58 and lots cheaper if you look around.

You may want a false floor - a board over underlay with underlay and carpet on top. It will reduce the annoyance to the people below. If you have next door neighbours separated by a mere partition wall check with a professional how to build a "box within a box" (a room made entirely of chipboard with NO rigid connection to the walls ceiling or floor). This is the ebnefit of the experience of the parents of the drummer in my daughter's electric band.

Plug the guitar if you can. Bugs (and woodies, and bottletops) are useless. They sound like dogs***, but you could try a tiny little Tandy (Radio Shack) voice lapel mic on a bed of blu-tak on the scratch plate. Martin Carthy does it so it can't be all bad. It is a more traditional sound than the modern scooped sound with treble added tizz!