No dog hair here but we have two cats; however I brew in the attached garage where they are not allowed specifically to minimize fur contamination. Ditto on the hot shower and fresh clothes on brew days, also brush my teeth and use listerine after reading here about another brewer who identified breathing over kettle and fermenter as a potential vector for infection. Sleeves stay rolled up, dipping hands frequently in sanitizer then wiping hands/forearms with wet paper towels. Can't be too careful.
Also, my wife knows full well no scooping of litter, running the vacuum cleaner, or anything else that raises dust until after the yeast has been pitched and the fermenter is sealed up.
Could you install an immersion chiller through the lid? Drop that on just before flame out to sanitize before switching on the coolant flow.
I too like to keep the lid on the fermenter once it's been sanitized, and transfer from the kettle valve with a short length of silicone tubing through a hole in the lid, and drape a sanitizer-soaked paper towel over the lid hole to minimize the gaps.