Definitive answer to the clicking or solenoid vibration...................the solenoid contact is held back by a spring and battery volts energize the contact to move against the spring to move over to connect. When the battery is weak, the contact still moves but the spring will override the lower magnetism because the battery is low and the contact springs back to then get magnetized again and over and over. The rapid back and forth action striking contact then breaking contact is what makes the noise.
Simple grease on terminal posts will stop all corrosion and last forever, unlike the store bought sprays. Messy but I have not had a single case of terminal corrosion since beginning to use it some 30 years ago. You may need to wipe and reinstall a new coat every 3-4 years or so.