Present Simple vs Present Continuous

I AM furious with John! He normally IS a very reasonable guy [,] but he IS HAVING some problems with the loss of his wife and IS TAKING OUT his anger on the group.

This is a fine sentence. Just add a comma after guy and omit the he before is taking out.

As was pointed out in a comment already, the use of the present tense is fine, too:

I am furious with John! He normally is a very reasonable guy, but he has some problems with the loss of his wife and takes out his anger on the group.

In comparing the time element in your two sentences, you can see the sentences have a slight difference in meaning. I'd be hard-pressed to explain what that slight difference is, however! By providing a larger context you would likely bring the difference, if any, into better relief.

First, let's assume that the first two verbs are conjugated in the present tense, since that is a reasonable base assumption. So we have:

I am furious with John! He is a very reasonable guy...

Easy enough. The difference between the semantic implications of the second part of the sentence become more evident now:

... but he has some problems with the loss of his wife and takes his anger and bitterness on the entire group.

The (simple) present tense implies that it is a habitual action; that is a part of who "John" is. In the sentence above, John is a man who has problems with having lost his wife - for example, he never got over it and has been miserable ever since. Being in such a disposition, he takes it out on his friends -- that's just who he is.

... but he is having some problems with the loss of his wife and is taking his anger and bitterness on the entire group.

This scenario is much more sympathetic to John. The present continuous implies that the act is ongoing -- and, key here, that it may end or change. Thus, in the above sentence, we can assume that he has lost his wife (relatively) recently, and he is currently having problems dealing with it. As a part of coping with the grief, he is taking it out on his friends -- but this isn't normally the case (perhaps he is usually more agreeable).

Your verb tenses are good. What makes this sentence a little strange is that the word "problem" trivializes John's grief. If you said "he is grieving the loss of his wife," it would sound more compassionate.