Delete photos from mobile device after synced to the cloud will they delete from there also?

Images that you have imported and synced to Lightroom cloud storage, which are shown in the Lightroom tab in the Lightroom Mobile app are available to you on any mobile devices, and via any internet browser at https://lightroom.adobe.com 

The device tab in the Lr Mobile app is just a view of your camera roll, and Lightroom does not sync images there unless they are imported into the app (either manually, or via auto import preferences in the app settings).  In the Lr app Device tab, imported images have a tiny 'Lr' badge.  Imported and edited images have a Lr badge with a pencil.  

If you have a "Lightroom" folder on your camera roll, that may have been created from Lightroom exporting images, but does not necessarily mean Lightroom is maintaining that camera roll folder through cloud sync.  For example, you could add images from another source to your camera roll Lightroom folder, and the Lightroom app would be oblivious to that.

I'm not sure if this is what you mean from your question above.  My recommendation: start slowly, delete one or two unimportant images from your camera roll which you believe Lightroom has imported.  And then check to make sure Lr has imported them.  I mentioned the Lightroom Web app website above, which works well as an independent way to confirm images are synced from your Lr Mobile app.