Hello!
I'd pay 300$ or less for the speakers, and not more than 100$ or 200$ for the earphones or headphones.
I would like them to be able to sound deep, even sub bass range. Don't know if I'm asking for sub bass and linearity of sound (like a studio monitor) AND asking for good bass is a contradiction. Generally hi-fi fans are all for bass boost and coloring of the sound. I have a hunch linear sound would be better for metal all around, since I read hi-fis can't go deep enough and overcompensate with a thumping, low quality bass. I have an old 2.1 speaker set I can't even find on the web, wouldn't post images of it ATM.
I would want to ask you musicians too and it seems obvious you would recommend a near-field studio monitor. I'm curious if far-field, general hi-fi might sound better or the previous. Unfortunately there's no speaker/hi-fi store in my area so I can't try any out.
I would be listening to metal sitting at the desk, or while doing chores around the house, exercising, etc. I would probably switch to headphones/earphones during if the speakers are near-field and I'm in another room, exercising, and the like. My room where my desk is measures like 10 square meters.
Until now, I have been looking at the Edifier R1280T. I live within the EU.

I'd recommend ones that are in your budget.

No idea for consumer speakers. I don't waste my time in that market anymore. Only studio monitors. Eve Audio SC204/SC205 would suit you well, but you will need an audio interface to plug them into.
Headphones, Audio Technica ATH-M50x, they also come in a wireless version now.

As for speakers, just remember that your room matters. Even if you go to a store and listen to 10 sets of speakers, you must consider the environment. In your 10 square meter space, bass could be bouncing all over the place, especially if you have windows or high ceilings with zero acoustic treatment or something like that. (Translation: it'll sound very boomy and undefined.)

We can't really recommend you anything because you just have to try some speakers and find out what sounds good in your situation. However, headphones, the ATHM50x are a good place to start.

One point easy to overlook/cast off when considering headphones for listening is long term comfort. The ATH M50s are top of the line when it comes to audio quality but are known for not being very comfortable. Unless you have a small head, they will press tight against your ears, I didn't like the feel even just having them on for a few minutes as a test.

Right now I own a pair of Sennheiser HD 579. They've got a bit of boost in the bass, as I feel that they over-represent the low end when I'm mixing. But are infinitely more comfortable to the ears and I feel like I can wear them all day long. The issue for me is eventually the band starts to press into my scalp, but that's much more tolerable so I really don't mind it.

Just the other day I tried on Beyerdynamic DT 770 and goddamn it was like wearing nothing at all. I'm definitely aiming to get a pair of these for my next headphones. Not as bassy as the Sennheiser I tried along with it (don't remember which model).

All three headphones mentioned should be in the $100 - $150 range, and are a great deal in terms of price/quality ratio.

I've owned a pair of ATH M50x's for about 4 months now. I had an issue with the top of my head getting a little sore for the first week but I tried a few adjustments and they've been super comfy ever since. I think I was trying to wear them too tight or something. I love them--I forgot how satisfying it is to listen to music on high-quality headphones. It's such a treat compared to the earbuds I was used to.