"Ben çok gencim."

Yes, that is correct. Evet.

Yes you can.

When a word ends with: k t ç p ( all the consonants in the word KeTÇaP).. And a vowel come after it.. K become ğ.. T become b.. Ç become c.. P become b.. Examples: Genç.. Gencim Kitap .. Kitabın Balık.. Balığı Yurt.. Yurdu

There is special cases like: saat .. Park.. etc

When consonants have a vowel after them, they change to the voiced version (I don't remember the exact wording or if it was vice versa, sorry). You can see what they change to in the accusative lesson.

Because of the" ketçap" grammar check it on YT

No, you must have the personal ending on "genç" which will have the form "gencim."

No, It's not right