Only a partial answer:
the Gimbaled Antenna Assembly Step happened and I wondered if there is a second antenna in case this step fails for whatever reason, and if not, what are the criteria applied when deciding whether backup systems are installed?
There is a second antenna, and it is part of the Gimbaled Antenna Assembly.
(And two back omni directional antenna which may have been removed from the design)
The second antenna is a small cup type antenna that provides a backup function.
the James-Webb has four (antenna): one at low frequency called “medium gain”, which consumes little energy, which guarantees permanent contact with the Earth and transmits the on-board parameters and the telemetry of the probe, another at high frequency called “high gain” which will allow the scientific data collected to be transferred, then two other antennas which will serve as backup because they are omnidirectional.
Since the launch, it is the medium-gain antenna which has ensured communication with the satellite, and the monitoring of its deployment. It operates in the S band, at 2.27 GHz
Although the above says there are two omni-directional antenna I have not seen any further reference to them.
JWST has a 0.6 m Ka-band high-gain antenna (HGA) as well as a 0.2 m S-band medium-gain antenna (MGA). Both are mounted on a common articulated platform, generally referred to as the HGA platform. The HGA platform can be articulated to point at the earth for any orientation of the observatory. The broad beam pattern of the MGA ensures that 40 kbps real time S-band telemetry is available with any visible ground station. S- and Ka-band links can be operated simultaneously and support all communications for commissioning and normal operations.
HGA and MGA, the latter also serving as a back up.
Some other back up mentions:
"Webb houses two HRGs, each internally contains two processors and power supply boards (2 for 1 redundancy) cross strapped to 4 gyros (4 for 3 redundancy). In the current architecture, one HRG is active receiving commands and providing telemetry, while the other HRG is in backup mode."
"Both the Power Control Unit and Telemetry Acquisition Unit contain completely redundant systems with one active while the other is in standby mode or off, completely." For the Solar Array Regulator, of the four converters, only one or two need be operating at a time with the other two on active standby.
Attitude Control Subsystem includes 1 internally redundant inertial reference unit
Backup battery mention:
Webb does have backup battery power, but the issue was drawing down the battery more than expected. That said, power output remained positive and Webb was "never power starved"... Webb engineers changed the solar array's duty cycles, and then double-checked the fix would be adequate.