Well my friend has 3 boys.
On all their doors... they simply put in a latch... those hook and eye latches on their doors.... WAY up high... at least 5 or more feet up the door... so that the kids cannot reach it.
That solved that problem, of their boys opening any and all doors.
Naps...at this age of 2, they often have nap and sleep tweaks. They are changing so much....
Make sure, he gets LOTS of runaround time in the MORNING... before lunch. Boys... are like wild horses... they NEED to exert themselves physically and get their yah-yahs out.
THEN, come home, have lunch. WIND-DOWN before nap... making everything keyed down and quiet. Verbally cue him that after lunch it is wind-down and nap.... my kids take about 1/2 to one hour to wind-down and key down, before nap. For example.
Do NOT do all of the nap prep... AT nap time. Do it way BEFORE nap time. Otherwise, nap time... will get too late.
Always wind-down and key down the child, PRIOR to the actual nap time.
DURING wind-down... keep everything quiet, not hyper, calm and even make everything dark if you have to. Close any curtains etc.
That's what I do.
Your son does still need a nap.
He is over-tired.
Which you know.
My son, is 4... and he still naps everyday, in the afternoon.
I do, what I mentioned.
He naps for 2-3 hours. And still goes to bed at night just fine.
If your kid is trashing everything in his room... remove it. All of it. Leave nothing in his room, besides basics. Tell him this.
all the best,
Susan