Interesting proverbs--thanks for sharing!
But the references to dentists with bad teeth and barbers with bad hairstyles are qualitatively different from cases of carpenters with broken doors and potters with broken pitchers. In the case of the dentists and barbers, the point is that they can only work on others' teeth and hair, and not their own. Hence, if they have bad teeth or hair, it is not a reflection on their own skills.
In fact, the observation that SirWillieTheWool is making is usually phrased as a fuller puzzle, wherein you go to a town that only has two barbers, and one of them has a great hairstyle whereas the other has a terrible hairstyle--and you are asked which barber you should let cut your hair. Suffice to say, you shouldn't go the barber with a great hairstyle.
Conversely, a carpenter and a potter are capable of working on their own homes and wares, so the analogy of the dentist and the barber doesn't extend to them.
The word for "sign"/"indication" is "Zeichen", not "Zeich". That said, I don't think your grammar is quite correct, but I'm not confident enough to correct that second clause.
"Alles" means everything; "alle" means "everyone" (and takes a plural verb). The final clause is not correct either, but I'm not sure what you're trying to say there.
In Russian we have an idiom "сапожник без сапог" (translates like like "bootsmaker without boots"). Rather means a joke about a professional worker who is potentially able but does not have an opportunity (money, time, etc.) to make something for him or herself. It is sometime ambiguous because on this words we (thanks to our history and culture, our people was always be in generally poor) imagine a man who work hard to make shoes for others - but his own foots are bare. We can also say it, for example, about a teacher who can't do anything with his or her own children, because he or she, first of all, their parent not a teacher for them. Anyway, in generally, it does not have some bad annotation like with dentist's bad teeth (or it only sounds such for me?)... but sounds in the same way to it.
it is quite logical once you get to it actually/know the genders/know which verbs/connectors invoke which case.
tho their table of endings seems to be somewhat broken, I am assuming it will be restored to what it was before.
I recommend you to screenshot/take a photo of the 3 inflection tables of german adjectives in Wikipedia, every time there is an inflection (of everything, not just adjectives, also words like welcher, alle, jede, die, ein ect) and fit the cases and the inflections to the sentence, it helps a lot to remember the correct adding (like "e" to schlecht in this example) and understand the cases better.
In this case it is plural, so the adjective has to have the plural ending "e." Even with an article it has an ending - ex. die schlecht"en" Zähne. Declension is difficult to learn. I find using the "strong and weak ending" method most useful, using charts as a quick reference when I need a reminder.
weil er schlechte Zähne hatte, bevor er beschloss, Zahnarzt zu werden
I suppose it differs from coutry to country and reflects how they study medicine.
Here, the differerence between a stomatologist and a dentist is like between a doctor and a medic: dentist is someone without higher medical education, assisting in teeth treatment or doing teeth prosthetics. There's also a teeth doctor, who doesn't know all the stuff stomatologist does and thus doesn't handle stuff that touches parts of mouth that are not teeth.
So, looks like "dentist" and "stomatologist" are different beasts and thus only "dentist" can be used here.