What does "coming down in stair rods" mean? Is it an English idiom? If so, in what circumstances can I use it?

Does "you were ever my best friend" mean "you were friends in the past, but not now"? What does "ever" mean here?

Yes. You are correct that she is referring to the fact that they were friends and now aren't. "Ever" here means at any time or at any point. So she's saying she's sorry that they were best friends at any point. Ex: Nothing ever seems to upset her. Ex: He doesn't ever eat his vegetables.

Which words were skipped after "I’d better"?

I guess he was trying to say "I'd better go."

What is the meaning of "filter down"?

The idiom "filter down" means "to pass or spread downward from an upper level to lower levels, as through a filter". It usually refers to information, resource, or communication. Ex "It always takes forever for any important information about the company to filter down to to me and the interns."

What does "stem from" mean?

To "stem from" means to be caused by, to come from, or to originate from something. Just like how a new flower bud grows from a stem, to "stem from" expresses that one thing is developed from something else. Ex: The wood damage in the house stems from the termites. Ex: Her love of music stems from her parents.

What does "coming down in stair rods" mean? Is it an English idiom? If so, in what circumstances can I use it?

Yes! The phrase "coming down in stair rods" is an English idiom. It means the rain is coming down very hard or it is raining heavily. You can use it when you see that it is raining a lot, as a way of commenting on the weather. Ex: It's been coming down in stair rods this week. I've had to drive through so many big puddles. Ex: We haven't had rain like this in a while. It's coming down in stair rods!