Welcome to the RobotShop Forum!
1. If you've cleaned the terminals on the Roomba and on the home base, then that is good. Can you tell if the home base's terminals are stuck in for any reason? They are spring loaded to accomodate the Roomba, but sometimes, if they get dirty or damaged, they can get stuck pressed down and may make poor contact. Also, after having cleaned the contacts on the home base and Roomba, do you notice if they are scratched and/or pitted? This may also contribute to the problem.
If there is nothing physically wrong with the setup, then it is likely either a defective motherboard, or a defective home base. Can you test the home base's output voltage? It should have 4.5VDC approximately.
2. It sounds like your control panel is defective. If you're okay with opening up the robot, we'd recommend you open it and remove the control panel from the top cover to clean it first. See the Ultimate Roomba Resource Center
for repair manuals that will explain how to disassemble the Roomba
(it doesn't show specifically how to replace the control panel, but it shows how to remove the top cover which houses the cotrol panel, and getting to the control panel is easy once you get that far). Otherwise, you may need a new control panel
. Please let us know exactly what model of Roomba 400 series your have so we can confirm if we have that particular type of control panel. We may have a few other options available.
3. The Discovery series scheduling remote never turns off if the batteries are installed; this is normal and is not a cause for concern. When you say you had to reset the remote, what was happening? Was the remote losing it's program, or were you just trying to get it to turn off? Normally as long as the batteries are left inside the remote, the programs should be saved. It's a good idea to leave the batteries in the remote until they need changing.
The last time you inserted the batteries into your remote and the Roomba wouldn't turn on, did you notice anything strange with the remote, or was it bahaving properly? Try making sure the 'window' on the front of the remote (small rectangle of transparent black plastic) and the cylindrical sensor on top of the Roomba's bumper are both clean and free of any buildup or debris.
Let us know what you find out or if you have more details. It's rare we see defective remotes.
For robots that have several things wrong with them, the cost and time required to fix them can start to rise quickly. You do have the option to trade-in your defective Roomba and get a 40$ credit applicable towards a new robot purchase. It could be used for a Roomba, or any of the other robotic vacuums