Is it possible to walk from Gatwick Airport station to Travelodge Gatwick, or other hotels in the area? Or if I need to take a bus, does the bus have rooms for luggages?
The travel lodge looks a little far away to me, however having recently dropped off my car for parking directly outside the Premier Inn I can confirm that both the Premier Inn and the Sofitel are onsite and easily accessible from the terminal by foot (I was using the North Terminal but it's also pretty easy to get between terminals on the inter-terminal transit shuttle).
Looking at google maps, it appears to be possible to go from the airport to the train station, and take a footpath North West towards Riverside Garden Park. Switching to google streetview, there is a footpath along the North East side of London Road which can be followed to the North West towards the Travelodge at the junction of London Road and Povey Cross Road. It looks like it shouldn't be more than a 10 minute walk.
My proposed route is shown here at a little over a mile, so more like 20 minutes to walk.
Travelodge runs shuttle buses. Fares are £3.00 per adult, one way and £1.50 per child (5-15 years old), one way.You will have to call up the hotel to let them know that you are waiting at the pick up point.
It is a long way from the airport, not a walking distance by any standards especially when you have luggage.
Getting a taxi would be cheaper if you are two adults or more but do not get a taxi from the airport taxi desk, call up some company like Alpha cars from Crawley instead and it would work out a lot cheaper.
I have never done it but doing some research if you walk to the corner of the Premier Inn Car Park and into the trees there is a small bridge over the River Mole 4 steps up and 4 down this brings you at the Travelodge Car Park . There is a local bus starts at 3 am for £1.60 each free if you have a bus pass OAP. Fastaway bus 100 stops at the North and South Terminals
I believe this would work and would take no more than 15 minutes from Gatwick North Terminal. Follow directions to Premier Inn London Gatwick Airport A23 Airport Way an easy 10 minute walk from the North Terminal which I have done many times. Going through the trees across the River Mole to the Travelodge Car Park could be a challenge if you have luggage and it is wet.