Hypnotizing a Big Daddy who has a Little Sister with her causes the Little Sister to run into the nearest vent. You can't harvest or rescue her while she's running or climbing in like this. (Possibly because that Big Daddy you hypnotized wouldn't approve)
I didn't find Bioshock to be too challenging. I beat it on Hard with Vita-Chambers turned off for the achievement, died once or twice trying to 'save' myself a Medkit when fighting a Rosie.
The biggest tip I can give you is to be professional and methodical in looting everything from the city under the sea. There's lots of first aid, ammo, and money lying around if you're willing to look for it; it's just usually not directly underfoot. Open every door, pry every crate apart, hack & shock every door, and there should be enough to get you through. It'll also find you a few more gene tonics and plasmids.
Also, the Big Daddy fights go much easier if you prepare the battlefield. Bouncers are easier to set traps for, because you know which way they're going to move (straight towards you!). Rosies are a little tougher. I'll pass on one dirty trick that I tried once to see if it worked (it did, viciously). If you have Proximity mines, fire them all onto a object you can lift with Telekinesis. All six of them you carry exploding at once will deplete a lot of that monster's health bar. That, plus loading your shotgun with Electric Buck and your Machine gun with Armor Piercing (if you have it) will drop them pretty fast.
If you get the chance, pump up your shotgun at the upgrade stations first. It's probably the most reliable death-dealer there is, and I used it for the whole game. Other early-game guns lose a lot of their punch, and the late game ones are all shorter on ammo or more circumstantial. The shotgun remains the staple hurt machine for the majority of the game in my hands.