> I don't need to ship my products. How can I unable the function "ship", so at the checkout customers have only the possibility to select "Pick up" ?
Have you tried removing all shipping methods? Then only the pick up will be left. Or better yet, instead of removing them, set some impossible requirements, like minimum price $10000. That's how I hide shipping methods on my store.
> Also, is it possible to have the "delivery" option? At specific days, time and region/city?
This sounds more complicated. Are you open to using a paid app (or a blend of apps) that would do this for you?
You may need to look into an app with advanced/flexible shipping rules.
By the way, you can change the labels of Shipping to say "Delivery" (in your themes language settings) and use the shipping solutions available. Still, specific days, time and regional availability is something that has to be researched further.