Is it permissible to give money to a poor woman at the men's mosque or at the women's mosque who is soliciting support, and is there any participation in sin or encouragement of begging in that?

There is no objection to giving charity to a poor woman near the men's mosque or to one who enters the women's mosque. Giving charity in the mosque is permissible according to the preponderant opinion, contrary to those who prohibited or disliked it. The hadith of Abu Bakr Al-Siddiq, may Allah be pleased with him, indicates the permissibility of giving charity in the mosque, as Abu Dawud dedicated a chapter to this in his Sunan. Asking for charity in the mosque is mildly disliked (makrooh tanzeehan), but giving to a beggar in it is an act of worship for which one is rewarded and is not disliked. Some jurists consider asking for charity in the mosque to be forbidden and giving to the beggar to be disliked absolutely, or they consider both asking and giving charity to him to be disliked because it is an aid to something disliked.