I am trying to correctly blazon the escutcheon of my family. Does anybody know if the following blazoning correctly describes the attached escutcheon?

Quatered first Sable, a bend or; second Gules, a lion rampant toward sinister or; third Argent, a cock vigilant gules atop a mount vert; forth Sable, dimidiated per pale Or, a fleur-de-lis azure – MECHTOLD.

This is very interesting, but what has it to do with Esperanto? Anyway, although I am not a vexillologist, from this glossary, what you have written sounds all right to me.

(sable=black, or = yellow, gules=red, argent=white, vert=green, azure=blue)

I know more the French blazoning terms than the English ones, but that sound mostly correct.
I'm not sure about the fourth quarter, though.

Since its separation is not in the middle, but rather on the first third, I'm wondering if it should be considered as a division per pale or as a pale (as an ordinary) positioned at dexter. The ordinaries occupy generally one-third of the field.
EDIT : In French we can say "adextré" (dextered?) to say that only the left third of the field is in one color.
When a charge is partially seen we say that is is "mouvant" or "issant". I don't know if there is a corresponding term in English, but I feel like the fact that the fleur-de-lis is only half-seen should be blazoned.

If you have an account on Reddit, you should try /r/heraldry/, maybe they can help you.

Mi provas blazonadi vian blazono esperante:
Kvarpartigita, en 1 zibela kun ora dekstra balteo, en 2 gorĝa leono grimpanta maldekstre, en 3 arĝenta kun gorĝa atenta virkoko sur sinopla teramaso, en 4 ora kun lilio enirinta de dekstra zibela flanko.

Aiwha:I know more the French blazoning terms than the English ones, but that sound mostly correct.
I'm not sure about the fourth quarter, though.

Since its separation is not in the middle, but rather on the first third, I'm wondering if it should be considered as a division per pale or as a pale (as an ordinary) positioned at dexter. The ordinaries occupy generally one-third of the field.
EDIT : In French we can say "adextré" (dextered?) to say that only the left third of the field is in one color.
When a charge is partially seen we say that is is "mouvant" or "issant". I don't know if there is a corresponding term in English, but I feel like the fact that the fleur-de-lis is only half-seen should be blazoned.

If you have a account on Reddit, you should try /r/heraldry/, maybe they can help you.

Mi provas blazonadi vian blazono esperante:
Kvarpartigita, en 1 zibela kun ora dekstra balteo, en 2 gorĝa leono grimpanta maldekstre, en 3 arĝenta kun gorĝa atenta virkoko sur sinopla teramaso, en 4 ora kun lilio enirinta de dekstra zibela flanko.

Thank you so very much. I just checked and yes, the term "issant" (or "issuant" or "issuing" ) is also known in English blazoning, for instance:

Azure, on a chief or, a demi-lion rampant issuant gules--MARKHAM, Notts.
Argent, a fesse gules, a demi-lion issuing therefrom sable--CHALMERS, Scotland.

What do you think of this blazoning:

Quatered first Sable, a bend or; second Gules, a lion rampant toward sinister or; third Argent, a cock vigilant gules atop a mount vert; forth Or, at dexter a pale sable, a demi-fleur-de-lis issant therefrom azure – MECHTOLD.

I also really like your Esperanto version. I am not sure, if "maldekstren" (toward sinister) would be more appropriate than "maldekstre" (at sinister).

Using your terms for the tinctures, but trying to stay as close as possible to the English word order, I would say:

Kvarpartigita, la unua Zibela, balteo ora; la dua Gorĝa, leono grimpanta maldekstren ora; la tria Arĝenta, virkoko atenta gorĝa sur monteto sinopla; la kvara Ora, dekstre fosto zibela, duon-lilio eliranta de ĝi lazura.

What do you think?

Yes, maldekstren sounds better than maldekstre.

I have just a relatively new interest in heraldry, so you should check your blazoning with someone how know better, but that seems correct to me.