What must I do, seeing that I have transmitted Hadiths that have been proven to me to be weak, and given fatwas that have become clear to me to be subjects of differing opinions, while believing them to be sound?

You must verify what is transmitted, for not every narration—whether authentic or weak—permits the derivation of a ruling from it. This is for the well-grounded scholars. And do not say what you do not know. If you inform someone about what has passed and find that you have erred, you should correct the mistake for them. The mentioned narration about 'Umar (may Allah be pleased with him) regarding not appointing Christians to positions of authority is authentic, as is the narration about Ibn 'Umar concerning the dislike of errors in the adhan. As for the issue of isbal (letting one's garment hang below the ankles), what you mentioned is one of two opinions, and it is appropriate to explain the difference of opinion if possible. If you are unable to correct the error, there is no blame upon you.