Can users edit their classifieds ?

Yes they can edit their own ads, you can use [adverts_manage] shortcode, this will display a page when they can see their own ads and edit them, see https://wpadverts.com/documentation/creating-ads-management-panel-adverts_manage/ for more details.

Great,
I saw that – but there is a small issue, when Admin is approving an add, and need to fix errors, – that Add appear under his name as Owner.

And the real Owner cannot edit it .
How we can avoid that?

This is a bug which we will be fixing in next release, it should be available this month.