Learn English – Is “Raining Cats and dogs” still used nowadays

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.

Mark my words, the phrase isn't all that outdated.

Nevertheless, I tried to think of other alternatives to this phrase, since that's what you asked for. I manage to think of a few:

• Don't say I didn't tell you (along with other variants)
• ...you can bank on it
• You can bet your bottom dollar
• I'm telling you...
• ...I guarantee it

Here are some excerpts where these phrases mean roughly the same thing as mark my words:

Don’t say I didn’t tell you: Two years from now, the GOP will officially split into two parties... (from a news article by Charles Ellison, 2014)

And you can bank on this: while crusades may start out as one-man crusades, if the idea behind the enterprise is good, soon you'll have lots of support. (from Magic of Thinking Big, David Schwartz, 1987)

We can teach all the right responses in the world but if we never role model them - well, you can bet your bottom dollar you won't see them in your children. (from a book on parenting by Eydie Comeaux, 2003)

I'm telling you, there's only one way you gonna get to Norlins now, and that's by cab. (from Old Glory: A Voyage Down the Mississippi by Jonathan Raban, 2011)

You can replace the bolded words with mark my words, and the passages will pretty much mean the same thing.

As for the currency of mark my words, an Ngram hints that its usage may have peaked about 100 years ago. Yet even in recent years it still dwarfs some of the alternatives I've mentioned:

I think the most interesting part of that Ngram, though, is the sudden spike in I guarantee it, which seems to coincide with Joe Namath's famous Super Bowl prediction, further discussed in this column.