this battery company should of been able to tell you after the first go around if that battery was any good or not.
if the cells look sulfated or the electrolyte is low or that one cell they said was weak was actually bad the battery will not charge up enough to run the lift. They should of found this out during thier initial testing and known whether or not it would hold up to run a lift. A simple load test on a loadbank tester would of told the story.
A fully charged 24v battery will read in the range of around 26 - 28v depending on the condition of the electrolyte and the cell plates.
Get a standard volt/ohm meter and set it to dc volts and read the voltage from the battery plug and see what your total voltage is. If you have a charger run a charge cycle on it and then let it sit for a few hrs and then recheck the voltage and see if it has increased any. If the battery is any good it will, if it has not changed much then there is more wrong with the battery and needs to be serviced, maybe even reconditioned IF possible. Just FYI, sometimes they aren't worth the reconditioning and it just pays to get another battery.
You can check the date on the battery by pulling off the cell connecting bar covers (if they are still there) and it should be stamped into one of the bars or poles. The serial number and date code should be stamped somewhere on it, usually on one of those areas.

Now as far as the slack light that is probably for the chains, make sure they are tight because if they go slack it will kill the lift operation generally until the condition is corrected. There should be some tension on them if they are adjusted correctly.
You can loosen the chain adjuster on the mast end of the chain and take up the slack in the chain(s) if they are loose so the slack switch will close and re-enable the lift's operation.

Before you do this though check the wiring to the switches or the switches themselves and see if there is any damage. They should be mounted usually at the chain anchor on the platform end of the chain mountings.

can you give serial number off the lift's data plate?