Conditions for the validity of Salaah

Performing maghrib Salaah half and hour after the time sets in

Q: I started praying Maghrib Fardh about 30-31 mins after Maghrib started (according to the calendar) and finished 41 mins or so after Maghrib started. I am not sure if I performed it on time as it was already dark outside. Do I make up the prayer or is it valid because my intention was to pray the fardh of maghrib? I also prayed 2 sunnah what about them?

Pigeon droppings in the Musjid

Q: I have seen scores of pigeons roosting in many Masjids.  They keep flying around and a lot of their droppings are found on the prayer mat, some dry and some fresh.  I had the opportunity to witness so much droppings continuously being made by the pigeons roosting in Jama Masjid in Delhi during my visit once, when I had gone to offer my Dhuhr Salaah with the Jamaath.

I require the following clarification.

1. Is our Salaah valid if offered at such a place full of pigeons' droppings, dry or fresh?

2. Do we have to clean only the spot of the droppings if they fall on our clothes if we are in a state of Wudhu or do we have to do our Wudhu all over
again?

3. Similarly, I have witnessed the same scenes at many Eidgahs in different cities during my travel, during Eid Salaah, where sheep/goat droppings/cattle dung are found all around dry or fresh, which the people responsible for the maintenance of the Eidgahs neglect to clean. Is our Salaah valid in such places?  

4. What is the Shariah Ruling on these kind of najis?