Don't put a 9-speed chain through a 6/7-speed front derailleur like that. It will work badly because the cage inner gap will be spaced to expect a wider chain. Random internet advice loves to get this subject wrong and say "it's just a front derailleur, it's simple when you think about it, blah blah blah." That is a good cue to tune whoever's saying it out. The usual problem that occurs with that mismatch is by the time you get it to shift down onto the small ring smoothly, you've opened up the low limit screw a ton because otherwise the outer cage plate is too far out, and the gap between the inner cage plate and the chain is now way too big when on the small ring. Issues with chain drop can result. Then when you go to shift the other direction, since the starting position of the inner cage is too far in, the outer cage will also be too far in for the middle and large positions, and you get rub against it in the smaller cog combinations on those.
Putting 9-speed chains on nominally 7-speed cranks usually can be made to work acceptably. The rings are spaced a little wider than ideal so some reduced shifting performance can arise.
The chain width differences are smaller in some speed generation changes than others, so there are examples where fudging FD/chain compatibility by one speed generation isn't such a big deal. Shimano/SRAM ten and eleven and are close like this for example. I know from experience that 9+ speed chains in this sort of derailleur work badly.
The other thing is that apart from their speed designation and spacing, Tourney and other similar cranks are made to much looser tolerances for runout in the chainrings. Narrower chains work with smaller gap tolerances at the FD and so they need less runout, not more. Between that and the above, there's quite a bit to recommend just getting a 9-speed crank.
All the rest works.