Is it permissible to sell an inherited bracelet, the Zakat for which has not been paid, for the purpose of spending on seeking knowledge, with a commitment to pay the Zakat later? And is it permissible to spend from its price on matters other than seeking knowledge?

Zakat is not obligatory on gold unless it reaches the nisab, which is 85 grams. Scholars have differed regarding the obligation of zakat on gold intended for personal use (jewelry), and the preponderant opinion is that zakat is obligatory on it. If your mother did not pay zakat because she was unaware of its obligation, or because she asked someone who issued a fatwa that it was not obligatory, then you are not required to pay zakat for the past years. However, if she knew of its obligation and neglected to pay it out of negligence, then you must pay zakat from her estate for the past years.