Can there be a half segwit, half normal Bitcoin transaction?
Yes, absolutely.

Every output can independently be a normal or SegWit output (when using P2SH-embedded SegWit, you can't even tell until it is being spent).

Every input can independently spend a normal output or SegWit output.

When a transaction has at least 1 SegWit input it's called a SegWit transaction.

When not all inputs are SegWit inputs, the transaction is still malleable.