Some REALLY good advice, so far!!!
First; yes, I would say that this is "normal" behaviour from a 2.5 year old. My daughter was 3 when I opened the daycare. She requires a lot of attention and positive praise, etc. She acted out, in slightly different ways, but still.
From day one, my kids's bedrooms have been off-limits to daycare kids. And I am very glad for that choice! They are encouraged to go and play in there if they need a quiet space or some downtime. They have some special things in there, and if they don't want to share them, they don't bring them out of the bedroom.