Drum brakes aren't any slower or quicker responding than disc brakes.
All new vehicles have to pass government N.H.T.S.A. braking tests or they aren't allowed on the road. Period.
Evaluating the braking systems of 10+ year old cars that have all been maintained and repaired differently, is...
A transmission shop will take a look at it for you for free, but be sure to get a second and third opinion, as a dishonest shop will immediately recommend a replacement of course.
Welcome to the Corolla Forum and Happy New Year to you too! :)
There are many threads and interesting video's here so feel free to jump in and take part. :)
How many miles do you have on your Corolla? Automatic or standard transmission?
It would be a good idea to disconnect the battery so it doesn't go dead.
If mice want to get into the exhaust, more power to them. I doubt they are going to chew on metal. :)
Well there's an old saying; "If it ain't broke, don't fix it". :)
With 71,000 miles it's possible the clutch just needs adjusting, (if possible with a Corolla). Call around to a few transmission shops in your area to inquire.