The Rock Warrior's Way describes a great technique for getting over a fear of lead falls that you can apply to this.
Basically, clip into an autobelay, Climb 5 feet up, and fall. Then climb 7 feet up and fall. Then climb 10 feet up and fall. etc etc.
It's all about building trust in the mechanical system. You don't trust it now because you're not used to it. Once you build up familiarity with the feeling of the system working perfectly as intended, you won't have that feat of the unknown when you're high up and feeling like you're gonna fall.
I once got told that the thing holding me back from climbing was that I didn't want it enough. Which pissed me off but I listened. I read 'The Rock Warrior's Way' (http://www.amazon.co.uk/The-Rock-Warriors-Way-Training/dp/0974011215) which was amazing, it literally gave me no fear. The downside was that in my no fear state I attempted a 5b trad grit route and fell way above my gear, twice, the second time my gear popped and I was inches from decking. Now I think a small amount of fear of heights is quite useful and I stopped reading the book....
a suggestion would be to read "Rock Warriors Way". It goes over quite a bit of the mental side of things.
I'd recommend this book:
http://www.amazon.com/The-Rock-Warriors-Way-Training/dp/0974011215
It teaches you how to identify irrational fear and put it to the side. As someone who recently went on the sharp end in quite awhile (I spent the last year bouldering), I had to reach deep into my bag of tricks for the bullshit slab smith rock throws at you.
Rather than claiming to know the answers to your personal problems, I'll point you to two resources that have helped me greatly:
Hope these help.
The classic book for mental game is the Rock Warrior's Way. Admittedly cheesy title, but some really good stuff in there.
https://www.amazon.com/Rock-Warriors-Way-Training-Climbers/dp/0974011215
So take what I say with a grain of salt because I only have two jumps in but this book was very helpful to me for climbing. Might be worth a look. https://www.amazon.com/Rock-Warriors-Way-Training-Climbers/dp/0974011215
Regarding fear of falling: mental training is a huge factor in climbing, not just for overcoming fear of falling but also for learning what to do about the internal dialogue.
I would HIGHLY RECOMMEND a book on this stuff that has helped a lot of people, myself included: Arno Ilgner's The Rock Warrior's Way. Definitely check it out.
I'd recommend checking out this book: https://www.amazon.com/Rock-Warriors-Way-Training-Climbers/dp/0974011215
Rock Warriors Way, or MasterMind both deal with the mental game.
https://trainingforclimbing.com/buy-books/jerry-moffatts-mastermind/
The Rock Warriors Way has really helped me get over some big mental blocks. https://www.amazon.com/Rock-Warriors-Way-Training-Climbers/dp/0974011215
Obligatory - Read "The Rock Warrior's Way" comment. (Seriously though, this is definitely addressed, and should pretty much be required reading for all climbers.)
http://warriorsway.com/products/
or via Amazon - https://www.amazon.com/Rock-Warriors-Way-Training-Climbers/dp/0974011215
As a beginner I would focus on technique (footwork) and sending weight (if you are not their yet). As you approach harder routes 5.11/12 and V5/6 you will absolutely need to train your forearm strength for pinches, slopers, and finger pockets.
Climbers tend to over-work "pulling" muscle groupings, and often forget their "pushing." This leads to among other things tendon stress which if gone untreated or balanced can cause severe injury over time (just look up pulley injuries). One way to train specifically in a well balanced manner is to work out the antagonistic muscle grouping (in climbers case the push muscles: Overhead Press, Bench Press, Tricep Extension, Push-ups, Straight-leg Deadlifts, Planks, ect...). The forearm exercise of opening a closed fist works out the opposite grouping of the muscles that close the fist. Since we as climbers overwork these muscles constantly as we grab and pull ourselves around, it's important to find ways to work out these antagonistic groupings.
I would recommend the Rock Warrior's Way for more information.