(1) non-logged user. the pencil icon in About section has no effect when clicked by non-logged user. The old design doesn't have such edit control displayed to users who cannot edit function definition

this reflects what we currently have in production, where non-logged users cannot edit function definitions, e.g. https://www.wikifunctions.org/view/en/Z10012

(2) the text on Details tab page (old design) - "Anyone can propose..." - is not present on Function page in the new design (not mobile-specific)

this is intentional. the text was only about test and implementations, not catering for function definition nor translations. and it was linking to an out-of-date page. we're testing an approach similar to Wikipedia where "anyone can edit" is implicit.

(3) the space between the lang icon and is rather tight.

+1. wondering if this might be Codex-related?

i would also add two minor issues to this task if you agree @Etonkovidova :

1. there is no space between the mediawiki header and the about widget, we can set it to 24px equivalent / @spacing-150
   
2. the bottom spacing between the widgets is inconsistent, we can set it everywhere to 24px equivalent / @spacing-150
   

Might be superseded by T349408.

I thinnk you have mistakenly assigned this task to me.

In T354978#9470098, @Jdforrester-WMF wrote:

Might be superseded by T349408.

+1!

(3) the space between the lang icon and is rather tight.

this is codex related, and it's being address here T356103: Natural space between icon and button text is lost in MinervaNeue