Nearest climbs in area

Not on the device itself.

You need to use strava segment search for that.  You can then send them over to the device and navigate to them.

So it is only possible when you know the names of the climbs?

No way to find climbs that you dont know yourself?

Segment Explore lets you find climbs on a map based on a number of factors including climb Category (1 - HC).  You don't need to know the name.  

Also possible to upload to the garmin edge when you dont have strava summit (paid version)?

I think if you "star" some segment in Strava it will sync to your garmin assuming that you have linked your Strava account to Garmin.  This happens automatically with Strava Summit.  I don't know how you would get segments into the Garmin except as part of the Live Segments app (or if you used Xert, which is another paid thing).  If you have segments in a starred Strava Route, that Strava Route (and the starred segments) will sync to the garmin and I think that is part of the "free" Strava.  Not sure if that will work with the Climb Pro app.

I guess it depends on what you want to do with segments on your Garmin.

Yes, you need to pay for Strava to see the Strava segments on your Garmin.