Where I am studying here in Makkah, I have seen many people praying without even removing their shoes. This looks quite strange to me. Is this allowed?
Yes, this is allowed. The need for praying with shoes on may generally arise in outdoor places like fields, tarmacs and pavements. Similarly, this need may also arise on bare floors inside houses. There is no harm in praying with shoes on in such places. One should just scrub the sole of the shoe to get rid of the big chunks of mud and dirt that may have accumulated there -- what little may remain is inconsequential.