What temperature should I wash at to kill bed bugs?

Bedbugs and their eggs can withstand temporary immersion in water.

To kill all stages, you should wash at 140°F (60°C) or tumble dry the cloth at the hottest setting for 30 minutes.

Clothes/items that cannot stand these temperatures should either be dry cleaned or freeze at -0.4°F (-18°C) for at least 24 hours.

Am I spreading bed bugs with the washing machine?

As long as you wash and dry at high temperatures there should be no problems. The hot water and drying process effectively kill any bed bugs and eggs. However, make sure to carefully handle infested items, ideally in plastic bags, to avoid spreading them to other areas.

Is it okay to use the shared laundry room?

Generally it is safe to use a shared laundry room, especially if everyone takes precautions. Carry your dirty and later clean laundry in plastic bags. Launder and dry your clothes on high heat to kill any bed bugs. Do not leave clothes or other belongings on the floor.