To describe what is actually happening right now, you use the verb form:
It is raining.
To describe the sort of day it is, you use the adjective form:
Today is a rainy day.
In your first sentence, either rainy or raining could fit, depending on what you actually want to say; "... because it is raining" indicates that water is physically falling from the sky right now, while "because it is rainy" indicates that it is the sort of day where rain is extremely likely to happen, but doesn't necessarily mean that rain is falling as you speak.
In the second sentence, only rainy fits because it is describing a quality of the weather, not an action.
Although synonymous in meaning, they're two different phrases created in slightly different ways.
"I'd like her to stop it". This is saying that you would like an event to occur. Other examples include, "I'd like a sunny day tomorrow" or "I'd like a super double pepperoni pizza please".
"I'd like it if she stopped". This says something else: you would like 'something' if an event occurred. The object is this abstract 'it' and the event is her stopping. Other examples: "I'd like Mel Gibson if he hadn't made so many bad films" or "We wouldn't like a pepperoni pizza if none of us would have it".
To answer the question as to which is better, I'll need to explain what I mean by "abstract 'it'". The 'it' is talking about the person's current situation. Example: "I'd like my current situation if she stopped", or "I'd like the state of affairs if she stopped".
As for which to use... this is more a matter of preference. You may notice that the two different constructs show emphasis on different things:
"I'd like her to stop it" puts more emphasis on the need that she stops what she's doing.
"I'd like it if she stops" puts more emphasis on the fact that like it if she stopped it.
That may have made it much more complicated than it really was, but, as is normally the case when one endeavours to work out how they've constructed their words, it was a journey of realisation for the author as well.
Best Answer
"Put down" means "placed on the floor, or on a table (or similar)". A teacher may say "put down your pencil" at the end of a test.
Instead you could say "closed her umbrella", or "folded her umbrella". I would also probably say that she "opened her umbrella" (but "put up" is quite clear). If you need an exact analogue to "put up" then use "take down".
("Put down" also means to insult, and for a vet to kill a sick animal so it has quite a range of meanings)