Some AC SSRs can be used for DC switching. Check specifications.
If you need to use 12V to trigger a 5V circuit, if both circuits have a common ground, use resistors to drop 12 to 5V. To be sure add a 5V Zener, or use a TTL comparator.
If there are separate grounds, use an optocoupler for best results.
What is your current rating- how much the 12 V source can make, and how much is the current load in 5V circuit?
I used PVG 612 "power" optocouplers, they need >3 mA primary current for the LED, and the secondary can hold up to 2A in 60 V circuit. Excellent devices, reliable to >110 deg.C
Bridge is the term used to convert AC to DC or to invert DC to AC.
It can be diodes, transistors, SCR's, Triac's IGBT's or Opto SSR.
It can be 1 phase or 3 phase or N-phase.
It is incorrect to use the term "AC-output SSR to output DC"
You either have an AC input or AC output or both.AC input and output.