Here you are what I would change:
My strongest characteristics are: proactivity, punctuality, the ability to talk to other people, making new friends and the easy adaptation to the more diversified situations. As for the weakest characteristics I would say: rashness* and the fear of failure.
* I find “rashness” to be a more suitable term for a letter of application if you are dealing with a job interview.
And, after the "fear of failure" I would definitely add something else, I think you should add something that shows that you are worried about your weakest characteristics, including hastiness/rashness (see: ...weakest characteristics I would say: rashness and the fear of failure, even though I find that fearing failure might be both strong and weak at the same time, as long as I can balance...
), the person who is going to read it might be a little worried depending on what side of the meaning they take into account.
I hope it can be of help.