I usually do a full window clean in the spring. I clean the outside, inside, screens and all the inside tracks and such. I use a special window "wooly" and squeegee with a rubber blade (this is important not plastic). Mine are similar to these from Lee Valley linked. I start by removing the screens - put them aside to be cleaned. I have a bucket ready to wash the glass. 1 capful dishwashing liquid & 1/2 cup of vinegar in a 1/2 bucket of water (approx 1 gal). I vacuum all the inside tracks (where the screens go) and then clean them all with a q tip. Spiders get in there and it's kinda' gross. I have casement windows so screens mount inside. I dip my wooly washer in the solution and scrub the window. Work the edges and corners well. For inside the window, I dip the wooly washer and then wring it out and clean the same. I place a towel on the floor inside to catch any stray water. After you've scrubbed the window clean - place the squeegee in the top left corner. Pull down applying light pressure. I wipe the blade after each pass. Keep moving across the window overlapping approx. 1" onto already cleaned glass. You can squeegee top to bottom or left to right - whichever works better for you. I wash my screen with mild detergent (I use Shaklee Basic H2 but I'm sure you could use a small amount of dish soap). I also add a little vinegar. Rinse them off good and allow to dry. I either rinse outside with the hose or in my walk in shower with the hand held shower head. That's my routine. I'd like to clean them in the Fall too but for now, once a year is what I do. Nothing nicer than clean windows :) Here is a link that might be useful: window washing tools

Wow, that sounds like a lot of work! I can't help you as I still have a little ol' Broan that uses mesh filters, which I put in the dishwasher. Is your hood working properly, meaning is any grease getting to the cabinets, ceiling, etc?

In my experience, most screens do come off. The faucets had to be assembled and you should be able to disassemble most if not all of them. It just may require a little ingenuity. If you absolutely can not get the screen out, you should be able to at least remove the business end of the faucet and backflush and/or soak the screen that way. Many manufacturers have lifetime warranties on their products and I would not be reluctant to seek a replacement if I had a malfunctioning faucet that could not be dissassembled to clean off a screen.

https://www.amazon.com/gp/aw/d/B00CP82FG2/ref=mp_s_a_1_5?ie=UTF8&qid=1482635128&sr=8-5&pi=AC_SX236_SY340_FMwebp_QL65&keywords=lcd+wipes&dpPl=1&dpID=410sUyH6CtL&ref=plSrch Zeiss makes great products for lenses and optics. These are alcohol and ammonia free as they should be. Be sure not to go in circles it can damage the screen. Go down in one direction from one side to the other.