Learn English – What are the differences between “sorry to keep you waiting” and “sorry for keeping you waiting.”

There is no difference in the example you have provided, but the second sentence (using by) is much clearer.

In some cases, it may cause ambiguity if you do not use by, for example:

You can avoid a lot of problems travelling on a train

This sounds like there are a lot of a problems when you travel on a train, and it is possible to avoid them.

You can avoid a lot of problems by travelling on a train

This sounds like there are a lot of problems which you can avoid if you travel on a train.

So there is some difference.

Also your last example, we don't say playing fireworks. We usually say watching fireworks, or launching fireworks depending on if we are just watching or actually participating. Also fireworks is one word.

By is optional in that case:

He celebrates the Chinese New Year by watching fireworks.

He celebrates the Chinese New Year watching fireworks.

Although they have the same meaning there is always a little more emphasis when we use by. In fact, in the case of the fireworks, using by makes it sound like a ritual or habit, whereas without by, it sounds like he is less concerned with how he celebrates.

As others have noted in the comments, it is often considered preferable as a matter of style to avoid reusing the same preposition in a sentence. This is not one of those situations.

I've been waiting 5 hours for him.

I've been waiting for 5 hours for him.

I've been waiting for him for 5 hours.

^ Each of these is perfectly fine in spoken English. The first sentence might be slightly preferable for written English.

Note that in informal American English, the most common way of rendering the statements in the original question would probably be something like:

I've been waiting for you for hours.

especially in spoken English.

(you might replace "for hours" here with "days," or "weeks," or "a really long time," etc., as appropriate)

I've been waiting ages for you is grammatically fine, but using "ages" with this meaning sounds a little stilted--especially to a younger American's ears.

I've been waiting you for ages.

^ this last sentence is not an option under any circumstances; when "you" is included, you cannot omit the "for" (unless you change "waiting" to "awaiting")