Hello and thanks for your inquiry. These are interesting and thoughtful questions that I hope I can help with, but certainly don't have definitive answers on everything. Below are answers to each of your questions.

What can I do to protect the tadpoles/fish and bees that might not be mentioned in the link? I know the mice will take the cotton balls back to their nest, and eventually won’t the water break it down and wash it into the environment and hurt the wildlife?

I don't have any direct experience with these tick tubes, but have used permethrin on clothing for several years. In my research on tick tubes, I found the Thermacell website to provide thorough answers on many of your concerns:

https://www.thermacell.com/faqs/product-questions/tick-tubes/tick-tube-product-guide

Their tick tube product guide also has some good info:

https://cdn.shopify.com/s/files/1/2537/3762/files/TRI_TCT_IBK_052219.pdf?12351

The Thermacell Tick Tubes also use permethrin, so their product info applies equally to these homemade tick tubes you wish to make.

Key points include "When Permethrin is bound to fiber it greatly limits the exposure of non-target insects".

"Will this product harm the environment?
When used as directed, Thermacell Tick Control
Tubes do not have any outdoor environmental use
restrictions. Unused product should not be placed
in indoor or outdoor drains. Unused product may be
discarded in the trash or offered for recycling."

I think given these points that the risk of exposure to aquatic environments is quite low, but as with most things, probably not a non-zero risk. As you say, eventually things breakdown and get released into the environment. It's just a consequence of long-lived chemicals. 

Below are links to several other fact sheets about permethrin; they cover many aspects of the chemical, but there are environmental/ecological sections that speak to your concerns quite well. 

Florida Department of Agriculture & Consumer Services

https://flsart.org/pdf/Permethrin%20QA_9-28-22.pdf

USDA Permethrin Fact Sheet

https://www3.epa.gov/pesticides/chem_search/reg_actions/reregistration/fs_PC-109701_1-Jun-06.pdf

National Pesticide Information Center

https://npic.orst.edu/factsheets/PermGen.html#env

If you want to be extra cautious, you can simply avoid placing the tick tubes near your wetlands. Mice have very small home ranges (around an acre or less), so mice pulling cotton from your tick tubes aren't carrying the material very far to their nests. Permethrin binds very strongly to materials and soil, so it won't simply mix with water and wash away into a wetland. It will also get broken down by microbes and sunlight.

If I put it on my clothes or my child’s clothes, when I wash it, will it find its way into the water system?

I think to some extent this is happening. But again, I think the concentrations are so low at this point that you're not appreciably adding to contaminant loads already happening from operations well beyond your control. If you read through some of the linked materials above, you'll see permethrin is used in broadcast sprays in agricultural settings and mosquito abatement programs using far greater volumes of permethrin, so the scale of things you're using permethrin for pale in comparison to many other commercial applications.

I want to protect it from the rain until the mice find it. Could I put it in a wooden Micheal’s bird house on the ground and just make sure they have rocks to crawl up into it?

I think that would work fine. There's a chance mice may actually just use the bird house to build their nest in, especially if there's already nesting material provided! To that end, that might be less ideal since the goal would be to have the cotton available to as many mice as possible, instead of just one pair and their offspring. I might just use a large piece of bark that you could probably find from a nearby dead tree and place it over the tick tube like a roof. Or place a thin board on top to save costs. 

On a similar note, I found this research article that tests the efficacy of these tick tubes. They found that the season is really important for how well they work, with colder months in the fall being the best time as mice are gathering warmer materials for their nests.

https://esajournals-onlinelibrary-wiley-com.proxy2.cl.msu.edu/doi/10.1002/ecs2.4155

Hope this helps!