"Is it usual to have this many guards at the gate or did the Rohirrim feel so threatened by the approach of the strangers that they called for extra men?"

I would think that during time of war, the gates would have a rather large guard force, that is the reason I think so many guards are there. I do not think three strangers would cause them to increase the guard.

"Gandalf just finished telling the others to "speak no haughty word", yet he immediately confronts the first person he meets. What's up with that?"

These men are like vikings. Haughty words to some may not seem so haughty to them. I think they expect a strong challenge to their rudenss from a friend. Soft words spoken to the Rohirrim would be met with even more suspiction than strong words, IMO. Gandalf knows this and responds accordingly.

"Is the guard really as hostile as he sounds, or is he just doing his job?  I can understand him suspecting that the grey-cloaked strangers are spies, but phantoms?"

The Rohirrim are fairly primitive. I think they are more hostile than normal due to the circumstances. They would be less so were the circumstances different.

"Are Gandalf, Aragorn, Legolas and Gimli really the strangest looking people the gate guards have ever seen? After all, these men see dead orcs."

Orcs are common, since almost every race has warred with orcs. Elves and dwarves are not common, since men, even moreso the Rohirrim, do not have dealings with the dwarves and the elves and vice versa.

You can tell that the Rohirrim have not seen an elf or dwarf for some time by the reaction of Eomer upon first meeting the three hunters. The Rohirrim stick mostly to their own and sometime have relations with the men of Gondor.