Learn English – The Times is a highly (respected or respectable) journal

I don't believe that "for many times" is grammatical in any variety of English.

^{(Well, unless you contrive an example where "times" is the plural of "a given time of occurrence" and there's an elided "of the": "Class times are listed below. For many times, you can click the link for more information.")}

You can use "for" with an ordinal: for the first time, for the 20th time, for the last time. Notice that it's always time, not times, in these examples. You can also use "for" with a duration: for 100 years, for the summer, for a long time.

"Many times" behaves the same way as "100 times" or "just a few times", and adding a "for" is not grammatical.

Mark has been to Hong Kong many times.
Mark has been in Hong Kong for two years now.

After Jessica failed 10000 times, she finally created a working robot.
After Jessica failed for the 10000th time, she tried a different approach.

Let's get the easier question out of the way first. When you are confused, you can use about or on, but I think about is the "safer" choice. Behold the ngram.

As for for the taking, let's say that you need a job, and I own a business. I might tell you:

I have a job opening. It's yours for the taking.

or:

I have a job opening. It's there for the taking.

Basically I'm telling you that you could get a job with me, if that option sounds good to you. I could easily substitute "if you want it" for "for the taking".

Looking at the ngram, we see that common ways to use the phrase include

• there for the taking
• free for the taking
• [possessive pronoun] for the taking (i.e., his for the taking, yours for the taking, etc.)

I won't say those are the ONLY way the phrase can be used, but an answer can't delve into every possible usage of a phrase. Those three are a good start.

There's also a relatively high number of

• compensation for the taking

but those come mostly from legal documents; that's not a wording generally heard in everyday conversation.