Only the first one is natural English. Personally, I would recommend saying:
Thank you for submitting your application.
It's a very natural style of success message.
"Thanks" isn't incorrect, but is a little informal for this sort of use. For a program for submitting application forms, it's probably better to be a little more formal, and so "thank you" is better than "thanks".
"Thanking you" is strange. If we thank someone, we will always say "thank you". It's a set expression. "Thanking" is only used as a verb for "to show appreciation", for example:
When I saw him, he was thanking her for helping him.
Also, we don't "show trust on", we rather "show trust in". In this context, rather than "show", "place your trust in us" would be more natural. To "place your trust in [someone]" means to do something that relies on the other person not to break your trust. For example, trusting you not to lose/misuse their application.
This isn't as common as just thanking them for submitting their application, however.