Nope. You need the 'of' since you have 'more' and a pronoun (what). However, it would be fine if you dropped 'more of'. In fact, "Do what(ever) makes you happy" sounds better to me.
No. You can "do less of something", "do the right amount of something" or "do more of something". Note, though, that when you specify the "something" or the "whatever", then you lose the "of".
Do less of what[ever] makes you sad.
Do more of what[ever] makes you happy.
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