The 1n5818 is a Schottky rectifier.
When the 555 output is low, the transistor pulls its collector within a Vce sat (~100mV) of its emitter. That pulls down the Adj terminal of the regulator, lowering its output pin to about ~2v. The rectifier prevents the battery from back feeding into the regulator's output.
Power in could be full-wave rectified, poorly filtered DC; cannot be AC.
An unregulated charger (like you describe) will destroy an SLA battery in a short time. A proper SLA charger MUST be regulated and have some of the elements as the referenced design, (but done better)
1. Current limited while charging a discharged battery.
2. Current limited until the SLA battery reaches ~14.5V.
3. Upon reaching 14.5V, the charger switches to a lower constant float voltage, about 13.2V and holds the battery at 13.2V for days to weeks.
4. If a load is put on the battery, discharging it below ~12.5V, then the charger should switch on (current-limited) until 14.5V is reached again...