1. Maybe things didn't go well at work.

2. Maybe situation didn't go well at work.
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(Does things' mean 'situation' here?)

3. You'll feel better.
4. You'll feel good.

5. You'll feel better.
6. You'll feel well.

7. You'll feel healthier.
8. You'll feel healthy.
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(Is 'better' from 'well' or 'good'?)

In a way, 1 and 2 mean the same. "things" is general. "situation" is specific. What situation? So they don't really mean the same thing.

"better" is a relative term. One can feel "better" than one did before, but still not feel "good" or "well" or "healthy". After the first three days of a cold, we usually begin to feel better; the symptoms are less severe. That does not mean we are "over" the cold yet or feeling "good" or "well" or "healthy".