The braid wire running to each feature controlled lamps is 5.4VDC power, it's not ground. So measure that the braid has power, if not check the playfield power connector J1 at the rectifier board. It might have some burnt connectors that need to be reterminated.

Download the manual from IPDB:
https://www.ipdb.org/machine.cgi?id=2355

Right here:
Alltek_LDB_TP13.jpg

In most cases yes, otherwise some of the feature controlled lamps that switch on and off will flicker.
In this case you need to connect a jumper wire from TP13 to backbox light door where the feature controlled lamp sockets have 5.4VDC. The braid wire at the "3 Up" lamp should be a suitable hookup location.

If the none of the LEDs you install in the feature controlled lamps flicker then you might be lucky enough to be using LEDs that don't suffer the flicker problem and won't need the jumper wire.

Notes:
Not having the jumper wire to TP13 will not cause LEDs failing to light up.
All this has nothing to do with G.I. which is a totally separate circuit and doesn't suffer from the same LED flickering issue.

As I remember, the playfield power plugs into the rectifier board at connector J3.

All new boards except the rectifier board by the transformer which looks original. I can see burned connectors in the picture. That's probably what your problem is.

Rectifier J3 is for the backbox. J1 is for the playfield. J2 is for the cabinet.