Simply backing down is used before the engagement starts. Bowing out means you cease to take part in the engagement. I’m sure there are situations were you could use either term depending on the circumstances.

“He saved the Labour Party which, less we forget, came close to extinction.” The second sentence doesn’t sound right. I don’t know the grammatical reason why it’s wrong.

“I don’t like winter because it’s very cold.”

I would put forward that in this particular sentence the correct term is “like-minded as we are”. You are stating a similarity. To use the second expression you would be comparing a similarity between a group to another person or group of people. “Italians are very expressive when they speak as are we, the French”

The first sentence is the correct usage. “I do exercise on occasion”, “ I take exercise very seriously”, “I get exercise when I walk to the store every day” if you say I do exercise every evening. You are trying to convince someone of the fact as if they dint believe you. Exercise is only quantifiable when you speak of specific exercises like push ups, bench press, sit ups are exercises.

They have just arrived at the winter house. It’s an expression used in many different context to indicate arriving at a place or in a certain situation. In your context it’s a place.