I bought a seemingly fresh (assuming 3yr old) Acer Palmatum (or Japonicum) and I repotted it in fresh earth (not Bonsai earth) and a bigger pot, with a drainage, some gravel.
The bark got a bit damaged in the process as the tree was fairly wet in that area. Though it has been 4 weeks now and it does not seem to have started sprouting leaves or recovering.

I have been watering the tree as much as possible without having the ground too wet or dry-out (I think). The tree is outside enjoying the morning and evening sun, without overheating during midday; no full sun.

I am afraid it is dying and I want to ‘rescue’/recover the tree.
How can I check if the tree is still ‘alive’ and what could I do/try to save the tree?

I will check if I can get a few images tomorrow.

You cannot check to see whether the tree is alive.
I know people claim that by removing part of the bark you can see whether it is green. That just prooves there is moisture. And you have just damaged the tree, so I would not recommend. But.. From the bark in itself you can tell a lot. But my crystal ball is not working, so a picture, yeah. That would help.

Hi leatherback!

The bark got damaged when I re-planted the tree.

This image is hidden for guests.
Please log in or register to see it.

This image is hidden for guests.
Please log in or register to see it.

This image is hidden for guests.
Please log in or register to see it.

I hope you can let me know what to do, to save this tree...

looks dead to me. Bark had turned brown/purple and is dried out, not shiny and "full"

leatherback wrote: looks dead to me. Bark had turned brown/purple and is dried out, not shiny and "full"

Thank you for the feedback.
I am pretty sad, I really was looking forward to my growing my tree.
I do still have four Hazelnut seedlings (about a couple weeks old now).

Any tips what to do, when I buy a new Acer Palmatum? And what should I specifically look for when buying another tree?
When I repotted the tree, I used lava / volcanic soil as a drainage layer and then reused the soil the tree had come with.
Maybe I did too much damage to the roots?

You did too much damage all over. The way that bark looks.. You must have been pretty rough with the tree.

I actually thought I was very gentile, except that bark that was lost during the repotting.
I have to be even more gentile. I'll check YouTube on how to report a tree and removing soil from the roots.

Bark does not just come off a healthy tree. So either you were rough, or you were working a dead tree. Bark on living trees does not just peel of.