That's all feasible with the Arduino but it's a lot of "stuff" for (apparently) your first project.
Take it one SMALL step at a time... Start by learning to read a temperature or humidity sensor, etc.
If your heating pad is AC, you can control it with a relay. A regular electro-mechanical relay usually needs a driver circuit, but you can get relay boards with the driver and relay. Or, you can find solid state relays that can be directly controlled by the Arduino. (A relay is an electrically isolated, electrically-controlled switch.)
A DC heater doesn't require isolation, but you can't directly power it from the Arduino, so you'll need a relay, transistor, or MOSFET, etc.
The same goes for the lights.
You "regulate" heat by turning-on the heat when you're below your target temperature and turning it off when you are above the target. ...Just like any thermostat, air conditioner, or refrigerator.
...at specific times of the day.
You'll need a Real Time Clock Module. These keep better time than the Arduino by itself, and they have their own back-up battery so the time doesn't get messed-up when you loose power.
Depending on what kind of user interface you want, you may also need an LCD display. i.e. Without an LCD display, you'd have to set the time-of-day and program your events with the computer.
Also is there a submersible sensor that I can have in a bucket of water and it would alarm me if the water ran low?
Look for a float switch or a liquid level sensor.