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(They) Both are tall and both have brown eyes.     (Good)

At best the preselected word order is ambiguous and confusing.

(They) Both are tall and both have brown eyes.     (Good, less good if "They" is included.)

At best the preselected word order is ambiguous and confusing (proof follows).

Thanks to F.E. for bringing the correct terminology into the discussion. To recap the issue of ambiguity, there are three functions which "both" can potentially perform:

1. quantification adjunct applied to subject
2. marker of coordination applied to two predicates (usually with different verbs)
3. marker of coordination applied to two direct objects or predicate adjectives

All examples in the question desire "both" to be interpreted as the first case, a quantification adjunct. But none of the offered options do so unambiguously.

If "both" is placed before the verb, it can function as either 1 or 2.

If "both" is placed after the verb, it can function as either 1 or 3.

In common usage, it shouldn't appear in either location. This provides maximum clarity.

They are tall and have brown eyes.
Can you sing and dance?
My grandmother is old and wise.

However, this doesn't provide emphasis. For that, including "both" can be useful.

Both... of them... are tall and have brown eyes.
Can you sing and also dance?
My grandmother is both old and wise.

In these examples, only the first leaves the meaning of the term "both" momentarily ambiguous, and its function becomes clear to the reader ("fixed") very early and actively. In the third example, "both" is fixed as soon as it appears, because it doesn't agree with the subject. For the second example, "both" would be unclear, even after the end of the sentence, so the emphasis on coordination has been provided in a different way.

All fifty of the the Miss USA competitors both are tall and have brown eyes.

All fifty of the Miss USA competitors both are tall and have brown eyes.

"Both" still doesn't modify the subject. If you wanted it to modify the subject, then

"Both" still doesn't modify the subject. To make this clearer, consider the awkward but grammatically correct sentence (of questionable truth)

All fifty of the the Miss USA competitors both are tall and have brown eyes.

or, less awkward

You must both sing and dance to be considered for this acting role.

If you wanted it to obviously modify the subject, then