Student or Learner
Can I use this sentence?
"How do you feel she is?"
'I feel she's asking me something.' [or any other verb]
'What makes you feel she is?'
Here, 'How do you feel she is?' would be possible, but less likely
For the more specific case, asking about her health, I agree with 'How do you feel she's doing?', though 'How do you feel she is?' would be OK too (in Br English at least), and if asking about her reaction to something you might well ask 'How do you feel she is with/over/about this?'