I experience lag in all Live based gameplay on Halo: Reach.
Remember that like any matchmaking in Halo: Reach, one of the players in the game will be the host, so even if your internet is great, if the other players have poor internet then you may experience lag(especially if they are the host). This is one of the reasons the game will sometimes pause to migrate hosts even if no one is dropping out as it sometimes decides that the host is too laggy.
With regards to Firefight mode - the issue is that unlike standard matchmaking, if one player is lagging particularly badly the game will sometimes stop and wait for a few seconds for them. If their connection doesn't recover sufficiently in the time then I think the game forces them to quit which is why you sometimes get the paused Waiting for Players screen.