In your first two examples:
e.g. "Each of these shirts is pretty", and "They each are great tennis players".
Only the the first one uses "each" as the subject of the sentence (and therefore takes the singular verb "is"). In your second example, "they" is the subject and the verb is the plural "are."
1. Am I right to state that the following sentence is correct: - "Each of them is a great tennis player" and this is wrong: - "Each of them are great tennis players"
Yes, both of these use "each" as the subject, so both should have a singular verb to match.
2. Applying the same strategy, I don't understand how these two sentences at page 204 are wrong: - "Each company has similar issues" and "Every company has similar issues".
Just in terms of subject-verb agreement, those sentences are not incorrect. The issue discussed here is an idiomatic construction. If you are going to say that two (or more) things are similar, then you need to construct that as a plural sentence - plural subject and plural verb. "each" implies that there is something distinct / different about each company - you're separating them out - so (idiomatically) it isn't appropriate to separate them out using "each" when what you're actually trying to convey is that they're all the same - they can all be lumped together (in terms of whatever similarity we're talking about).