The one who says the sentence (whoever is me) is the one who is surprised that the other person, in this case she, was happy to see 'me'.
So it's a clumsy shortcut for
I was surprised that she was happy to see me.
If you google 'surprisingly happy to see me' you can find other examples of this phrase. The writings that this phrase is used in don't seem to rise above mediocrity.
It would probably be better to write
She was, surprisingly, happy to see me.
This takes away the clumsiness and it is equivalent to
Surprisingly, she was happy to see me.
You have already answered your second question, because you need merely say
She was both happy and surprised to see me.
If you say
She was happily surprised to see me
this 'happily' can also refer back to the speaker and not to she, as in
Happily, she was surprised to see me.
In this case it's a shortcut for
I was happy that she was surprised to see me.
You're welcome and I hope this was helpful!