A baby corn is fine a a 10 gal. tank - but an adult will need a 20.
The lights can follow the outdoor light cycles. Many people follow that rule. Then there are those like me. I leave the lights on for the reptiles 10 hours a day in the winter and 12 in the summer. The only way to know for sure if you have the right wattage light is to get a digital thermometer. Indoor/outdoor are the best. Put the probe in the basking area and if it goes above 88 degrees you have it too hot. That area should be between 85-88. Too hot means you just need a weaker bulb.
Your baby should be able to submerge itself in the water. Get a small tupperware bowl - or better yet - one of those ceramic pet crocks - which will allow for total submersion.