Learn English – “a couple OF times” or “couple times”? “a couple of minutes” or “couple minutes”

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.

Wait a minute

This is pretty generic and informal and is often used idiomatically to ask someone to pause what they're doing until told to continue. It doesn't usually mean exactly a minute, any more than "wait a second" means exactly a second.

Wait for a minute

This sounds like something that you'd find in directions/instructions:

Wait for a minute and then add the oil to the pan.

In this case, it more often means exactly a minute, though it's not required. The exact timing here is more important. You would be unlikely to find native speakers who would use this in the idiomatic manner of the previous example.

Now, does this carry over to numbers of minutes greater than one? Sort of?

Wait five minutes

Will usually mean five actual minutes... otherwise, why bother specifying the number. As mentioned before, "wait a minute" doesn't really mean a minute... one could end up waiting for a few seconds or several minutes in the end. Even in directions, you'll often find they've left out the "for" in this case:

"Remove from stove after one minute of boiling and wait five minutes before topping with granola."

So what about with the "for"?

Wait for five minutes

This isn't wrong. Using it this way is not ungrammatical at all, it's your choice. It's not required and you will definitely find examples of it, particularly in more professionally-written instructions:

Slide the whole skillet under the broiler, and wait for five minutes, until the crumbs are golden brown and the salmon is done.

This is also discussed, though not to a great degree, in Cambridge Dictionaries Online:

Wait means ‘stay in the same place or not do something until something else happens’. We can use it with or without for:

• Put a tea bag into the cup, then add water and wait (for) a minute or two before taking it out.
• I phoned the head office but I had to wait (for) five minutes before I spoke to anyone.