What should I do with the beer and the sign?
Read the sign.
'Take the beer under false pretences'.
What should I do in the bathroom with the duck and the large jar?
The duck is a 'lame duck'.
Open the door. (When is a door not a door?)
What should I do in the closet?
Get comb, 'search closet with a fine-tooth comb'. Get all.
'Give skeleton the deep six', 'empty baby out with bath water'.
What is important about the paper in the Public Square?
The paper lists the local laws; there are 9 out of 10 here.
'Possession is nine tenths of the law'.
What next?
Get horn, 'beat around the bush' to find a laurel branch. Get it.
'Wash dirty linen in public' then 'put blessing in disguise'.
How do I finish?
Go back in and bless the Mayor, then 'toot own horn'.
Give decree to the Mayor, then 'rest on my laurels'.