Welcome to the Org, Emren.

In regard to your second question: yes, there is a search function, but TosaInu disabled it for junior patrons because it brought the forum down on a regular basis. Also, the search function is as yet unable to track the topics of the old forum we abandoned a month ago.

Regarding your kings: princes tend to have about the same stats as the king has the turn before they mature (doesn't matter if the king is their father, uncle or brother). High influence kings get better heirs. However, there is a bug or feature whereby high command kings get zero star sons. Therefor some players recommend never getting your king above seven stars.
The starting influence of a king is faction dependend: Byzantine kings always start with great influence and therefor get great heirs. I haven't played with Hungary yet, but my guess is that they have a standard low influence and thus will get unimpressive heirs.

Getting fine heirs is an art in itself. There was beautiful post about this some time ago, but because of the search function-troubles I cannot find it. The essence is that figthing kings get better sons and are less likely to get nasty vices. So keep them in the field even if you have better generals available. This is also a great way to improve your influence. Heir improving is a low proces, but try for example the Viking campaign. You'll be amazed at how fast you can get a royal family of military geniusses.