I don't know if it's fair to say "strongly recommended" - while it's certainly acceptable to use -ею as the pronoun or certain adjectival endings, I've always been taught that it's an older form of the instrumental feminine ending and it definitely doesn't seem to be as common as ей.
I was talking about the form of the pronoun, not adjectives, though this also applies to the latter. The problem is, that as ей is also the form of Dative, in certain contexts you won't be able to distinguish the meaning if you use ей for both. For example: Это сказано ей. Does it mean 'It is said to her' or 'It is said by her'? You can't tell. Therefore for the second meaning you should say 'Это сказано ею'
The problem is, that as ей is also the form of Dative, in certain contexts you won't be able to distinguish the meaning if you use ей for both.
It's not that much of a problem, considering that the ambiguity only arises in very specific context which can easily be avoided by rephrasing the sentence. Most of the time there's no confusion, so native speakers habitually use "ей" for the instrumental.
"Как вы думаете" is a parenthetical phrase. It's doesn't directly relate to the following sentence, and is simply used as an injection. "Как вы думаете, сегодня пойдёт дождь?" - "What do you think, will it rain today?"
"Что вы думаете" is an integral part of the sentence when you ask about one's thoughts about some subject. It's usually used with prepositions "о" or "про" (both mean "about").
A good way to distinguish them, is to try and throw it away. If the sentence still makes sense without it ("сегодня пойдёт дождь?" has the same meaning; the only difference is style) then it's "как вы думаете". If without it the sentence falls apart ("о ней?" is meaningless on it's own) then it's "что вы думаете".