Those look just fine. No problem - plenty of life left in 'em. Plus, if they fail, it isn't like you are falling 2 stories.

Yes, and in addition to creating a joint that is attached on only 2 sides and can flex properly, it will also solve OP's problem of too much sealant oozing out while drying.

By using less sealant, properly adhered with a backer rod, all of it will perform better.

there it is - remove the handle with a hex wrench (allen key) at that set screw here. After that, you can probably just pop off the rose using a flat blade screwdriver under the edge, and you'll have access to the main fasteners that are holding that lock in place.

Yes. Need more pictures from all directions of this side of the door handle, and the other.

I'm all about these kind: Men's Solid Crew Lightweight Lifestyle Sock.

While sure, big thick colorful hiking socks have their place...but for city folks who go to desk jobs, these are your day-to-day excellent work approved (black) and also casual (gray) style that can work with both your brogues and your sneakers.

Warm enough for all but serious northern winters, and light and breathable for Spring and Fall. For summer, switch to cotton though.

I watch all year for sales, and can find them more in the ~$13 range.