Is it a condition that the meat of the Eid sacrifice should be distributed in order for the sacrifice to be accepted by Allah?

The poor and needy should be given from that sacrifice , even if it be a small portion,(One kilo for example) as raw meat so that they would cook it the way they want. The remainder of that sacrifice is either distributed, eaten, or offered to people in a banquette .

A man who was on travel prayed Duhr as four Rakhas upon leaving Tafilah heading to Amman. However, on his way to Amman, he prayed Asir as two Rakhas (Shortened). Is what he did correct from an Islamic perspective?

Should the Zakah(obligatory charity) giver tell the poor recipient that this is the Zakah of his money?

No, he shouldn`t. But, the Zakah giver should make an intention in his heart that it is the Zakah of his money when he gives it to the poor.

Is it permissible to delay menses by using medication in order to fast Ramadhaan?

It is permissible for a woman to use medication in order to delay her menses so that it becomes lawful for her to fast Ramadhaan, but it is better that she doesn`t do so, and taking the medication is prohibited if she, or her doctors know that it involves a risk on her health.