I am going to install a trap on the washer drain pipe that is connected to the main sewer line. The washer drain runs down 2 feet and across 3 feet. Will the trap work correctly if it does not have a tight seal where the drain hose meets the drain pipe. Will I have a problem with water backing up and out of the pipe?

I hate to rain on your parade but your plan isn't going to fly unless you vent it.
I don't know what you mean by this, "Will the trap work correctly if it does not have a tight seal where the drain hose meets the drain pipe.?"
If you install it the way you describe the washer discharge will pull the trap seal out with the discharge and you will be left with a direct open connection between the sewer and your home. What you can do is come off the sewer line with a vent, then a trap with a 36" stand pipe to pick up the washer.

Good luck, Tom