Another thing, would I have to pay $1 each time I send something (after I spend my three free sends), or this is $1 paid for lifetime use?
The second. You only pay once a $1 for lifetime use. This way the app is free to receive and remove a potential friction on the recipient's side.
...having Crosspass app as closed source and not really knowing what is happening under the hood... this is not exactly my cup of tea.
I can let you (and anyone with a decent status on BitcoinTalk) review the source code, provided you report back in this thread that here are no intentional backdoors or data leaks. Also, by end of this year, I will have it reviewed by a reputable pen-testing company. Once Crosspass gains enough installs to have market advantage over any clone, I will open-source it.
I really don't understand how Crosspass is related with Bitcoin

One's finances should be private.
Every time you post your bitcoin address, or email it, mass surveillance associates it with you and records how much money you have received. That's why ZCash went to so much trouble to hide the identity of the sender of crypto. By sharing a Bitcoin address via Crosspass, you become essentially anonymous. Watch a 5 minute Doodly video:
https://www.youtube.com/watch?v=NK-P_g6gKlI