Hi Thierry, 

Let's look at your first question, "stick" versus "paste" as verbs.

When we apply a layer of adhesive (e.g. glue) on a surface before affixing it to another surface, we say that we are "pasting" something. On the other hand, when we try to attach something to another surface with some kind of fastener (e.g. pin), we say that we are "sticking" something. 

"Stick" as a verb has other meanings as well. For example, children "stick" their tongues out when they are caught misbehaving. We also use "stick" for the act of inserting an object into something else. For example, a mother may stick her finger into a bottle of milk to make sure it is not too hot for her baby. 

In your sentence, "sticker" is a noun, and is correct. A "sticker" already has a layer of adhesive applied, so the user just needs to apply a gentle force on the "sticker" to press it against a part of a mobile crane.

Just as a side discussion to share a little more: 

"stuck" as an adjective means two objects that are difficult to separate (e.g. His leg was stuck in the drain.) while "pasted" as an adjective means something that is already affixed to a surface using some adhesive agent (e.g. glue, paste).

You may have heard of the phrase "stick around". This is an informal phrase for someone who is waiting around an area. On the contrary, we never use "paste around" for the same meaning.

So as you can see, the choice between "paste" and "glue" depends on your context and intended meaning. 

I hope this helps.

Hey Thierry, between those two words, I also would have chosen 'pasted' so that I didn't repeat the word lemma (group of words with the same base) 'stick' (i.e. inspection stickers STUCK). I would say that the word 'paste' used in this context is more British (at least, in my experience), but everyone would understand. 

1) 'kindly update all inspection stickers pasted on/to the mobile cranes' (you can also write 'pasted to' without much change in meaning)

2) 'kindly update all inspection stickers attached to the mobile cranes' (this is more generic as 'attached' doesn't have to be with a glue or paste)

3) 'kindly update all inspection stickers affixed to the mobile cranes' (this is also generic and is a synonym of 'attached')

4) 'kindly update all inspection stickers adhered to the mobile cranes' (in this context, 'adhered to' also has the same meaning as 'pasted on/to')

Hope this helps!

I agree with the other comments. Keep in mind that stick can also be used as a noun (feel free to google images of a stick). 

If you removed the word "pasted", I would understand it to have the same meaning.