Hello again,
It's a bit hard to tell these apart because ADONIS is also a PERMANOVA. It's even in the name!
adonis: Permutational Multivariate Analysis of Variance Using Distance Matrices 
Without getting into the underlying math, these are very similar tests and let you measure how different are groups from each other.
Yes, you can do this with both the basic permanova and the distance permanova (adonis) test.
Yep, and sequentially in the order of your choosing!! 
This is the true power of the adonis test, and what differentiates it from a basic permanova. When the distance matrix is partitioned, to compare 
vs 
or Room1 vs Room2, you can first 'pre-partition' it to remove variance (statistically control) factors you are not interested in!
So you can partition out the chicken variance so you can measure the effect of the room, like this: adonis(~chicken + room)
Or, you can partition out the room variance so you can measure the effect of challenging your chickens! adonis(~room + chicken)
Even better, you 
control for one and test the other
It can even do interaction effects!
adonis(~room * chicken)
To make use of the full power of this test, you need enough blocking in your study design so the variables in question do not confound each other. Based on what you have told me so far, I think the ADONIS test is a great fit for your question, as it should help you remove variance due to room and let you focus on variance due to the chicken challenge!
