what about that black stuff do I need to buy and put it back ? What is its name

What is its name

Likely #15in. felt, often called tar paper.

Absolutely not needed.

See Building Science Corp. here.

Of course, this may be the only part of the basement wall properly insulated. But might as well do it right.

I bought this one as it was the thinnest option available at HD near my house

Durofoam EPS Rigid Insulation 96Inch X 48Inch X 3/4Inch

they say install it using 2x2 furring strips -that leaves an air gap behind the drywall and I think this is to run wires. Now do I need that gap there ? I need to save room behind my toilet sit otherwise it won't fit.

So the plan is to cut 3/4 thin strips from a 2x4. I can't find it precut this days but the furring I removed was exactly that size so you can imagine how old it is -like 60y. The strips will be flush will be flush with the insulation and then the drywall will come on top of it. If needed probably I could leave like 1/4" air gap there by either using 1" lumber or cut it of a 2x4 with the table saw

Now the question is, if I cut my furring strips from a 2x4 pressure treated stud will the exposed side (now untreated I guess) be prone to rotting even it that side is facing the drywall (so it is away from the wall which I assume is the source of humidity?

The other side of the drywall will be tiled and it is dry since it is behind the toilet this is not a shower wall

Glue the Durofoam to the foundation wall, as per this Durofoam video: https://www.google.com/search?client=firefox-b-d&q=installing+durofoam+video#kpvalbx=_QZVSYaDSAeuW_Qbo76XYDg24 The furring is being used to hold the drywall screws/nails, so you'll end up with an unnecessary gap in any case. Furring comes in 1x4x8 pressure treated, so there's no need to cut 2x4s. If you can, support the furring without penetrating the Durofoam. Not much insulation value, but it should reduce the inward moisture drive and reduce condensation on the concrete wall.

If you have a standard toilet, you could consider buying a new toilet designed for walls that are closer. Standard toilets are 12" and they make 10" as well. would save you a couple of inches if needed.

I both this from Costco almost an year ago https://www.ovedecors.com/pub/media/amasty/amfile/attach/IM_Stan_Tuva_2020-05-06.pdf I did not realize there could be a problem with this

Not sure how bad it will be to have just 3/4" foam insulation in that corner

Not sur if I can find a similar one (smart toilet with bidet and no tank) within the same price range (700CAD)

As long as you have an adequately sized dehumidifier in the basement, I wouldn't be concerned about excess moisture.

That wall will get tiles over the drywall so I dont really know how much humidity will get through. I am more worried about what is going to build up behind that drywall as I have a window above this niche where the toilet is going to be installed. I could use tedgua4d over drywall and the thinset and the tiles will be 24xx24 This is a bathroom with a fan set to start automatically as soon as the humidity reches a certain level n