Yes please!

Like maybe have "groups" that are only for aligning purposes? This way I can keep them in separate layers. Otherwise I have to group-> align-> ungroup -> manually select and move objects to each of their original layers.

I wish there was a way to parent objects, link them like grouping, but independent of layer order.

This is done in 3D software and notably in After Effects with the pick whip.

Sometimes, especially in work like illustration or making assets for animation, things that are meant to be grouped together aren't necessarily all directly on top of each other.

A feature like this would greatly expend on Illustrator's abilities and keep it competitive in the future.

Grouping objects on different layers for moving etc, but keeping them in their original layers.

For example: I have a number of objects that are arranged nicely to each other, but I want to move them to the centre of the artboard. So I should be able to select all, group, and select the center button, then release them and not have their layers changed.

I mean, leave groups as the are, per layer only. But add another way of managing sets.

I would not call this 'Groups', but 'Named Selection Sets', like 3DSMax does it:
https://knowledge.autodesk.com/support/3ds-max/learn-explore/caas/CloudHelp/cloudhelp/2017/ENU/3DSMax/files/GUID-93B01505-5474-43D6-8B83-810BA0157A9B-htm.html

This would be amazing.

Like in Photoshop, I'd like to link paths (like PS linking layers) that does not group them together, compound them, or reorganize them within my document. I just want to be able to select one piece and have it select the linked paths at the same time, even if they are alternating with other paths within my layer.

(See Image: Shadows alternate level on layer, flat–shadow–flat–shadow–etc. I would just like to select those shadows all at once, while also having the same gradient within other non-linked paths).

Would like to be able to group items on different layers while keeping the group intact. This would be especially handy when using the Esko cut software (cut layers). With this option, I would be able to group artwork with a die to more easily impose art across various sheet sizes (across different documents or within the same document with different artboard sizes).

Grouping items in Illustrator means it looses all the layer information. While sometimes this is ok, or doesn't matter, it's frustrating at other times - could an option be introduced to 'keep layer information when grouping' (in much the same way that 'paste remember layers' works)?