Yes, your costumes and props are just as vital as those belonging to the other companies, and yes, the dressing room and backstage space is as much yours as everyone else's. But, as we have gradually been discovering, it is so very necessary to respect other peoples' property in those areas. To see spotless replica uniforms and hats lifted from their rail and strewn across the floor and to have fruit juice spilt over electric plugs and surfaces that are to be used on stage is not a nice thing, as I'm sure everyone else will agree!
Litter, too, makes things very difficult backstage - clearing it off props eats into companies' get in times, and rooting through it to find missing items takes up time in the dressing rooms!
It might be a good idea to buy a rail and label it to keep others from moving your stuff around or dumping it in corners. Also, assign a space backstage for your props. There's very little that can be done about littering and spillages, but as long as each person ensures that the spaces they use are tidy as they leave, life for other performers, and the lives of the venue's staff, will be made much easier!