"I have never been sick."
Translation:Ben hiç hasta olmadım.
The -lAn suffix carries the meaning of becoming and turns an adjective or a noun into a verb (yaşlanmak, sinirlenmek). -lAş (iyileşmek, kolaylaşmak) works in a similar way. I don't think you can use it with any word, sort of like you can't use -ify (solidify) with every word in English.
No, unfortunately you can't produce verbs that way in every case, so the examples you give are wrong. However there are a few verbs that are formed following the same ending. A couple that I can think of are:
Paslanmak: to get rusty
Yaşlanmak: to get old
Your examples would be güzelleşmek-to get beautiful and acıkmak-to get hungry.
In Turkish we haven't got perfect tense that's why you dont understand difference