How do I mount rear derailleur on frame without separate hook part? (Language barrier warning! Please edit if I used wrong english terms)

After few years my generic rear derailleur started to fall apart, so I got new shiny one aaaand I have no idea how to mount that thing. Video tutorial authors seems to consider this an obvious task, with which I have some problems as my hardware seems to lack that hook thing to which you mount those hellish spiny devices with french sounding name that eludes my browser thesaurus.

I have no idea how half of those parts are called in my language, let alone in english, so I made pictures! Also, I never replaced this part, treat me as newbie.

Right next to wheel nut I have this hook thing pointing straight down.


And here is old and new:

Old one was Shimano Tourney that came with bike, new one is Shimano Acera. Gearing is 7x3. (In my region choice is between no-name trash, pro-parts with pro-price and variety of Shimano, so I am pretty much stuck with this brand)

Old one was attached with this:


As it stands now when I just attach new one directly to frame, which is quite bothersome when its all coiled up, it can freely rotate around mounting screw (seems like feature) but nothing blocks it from going up (and screwing itself into cassette, which seems bad).


Can i just mount it and proceed with servicing?
Do I need some kind of adapter?
Or maybe it is not compatible?
How do you call that sideways regulation screw (with blue paint) near
attachment point?
Should this contraption go up on its? Is its position regulated by
chain tension?


(Yes, I am replacing all cables as well)
(Yes, I still have some cleaning to do)
Some Tourney derailleurs have a sort of extension ('hook' as you call it) that your new Acera derailleur does not. That is fine. The Acera derailleur simply bolts to the derailleur hanger on the frame (the piece if metal pointing downward in your first picture)

Your new Acera derailleur has what is known as a 'B pivot' which I believe the Tourney derailleur lacks. The B pivot is located where the derailleur bolts to the frame. The entire derailleur rotates around the B pivot, the B pivot spring tries to rotate the derailleur anti-clockwise underneath the cassette.

(There is also a 'A pivot', around which the jockey wheel cage. rotates.)

The blue painted screw you mention is the 'B screw', which is used to adjust the position of the derailleur about the B pivot. It is likely that your B screw simply needs to be adjusted properly so that the derailleur is moved away from the cassette

You can look up instructions on how correctly the adjust the B screw. Park tool has a good video on derailleur adjustment which includes B screw adjustment.