I think 3 months is a bit too young for a swim in the pool. Try baby-friendly swim schools like Little Splashes. Little Splashes uses a heated mineral pool. You can try their trial class to see if your baby likes it first.
Better not. The chlorine amount is too high babies below 1 year old. If you wanna try bribing your lo to swim, you can try babyspa or various swim school ( where they have heated pool and lesser chlorine treated water )
Adult pools are usually quite cold. I tried bringing my baby once and she shivered a lot and cried 😥. You can get a thermal swimsuit and try it out briefly to see if your baby is OK? Have a towel ready also.