Is it ok to start the engine without skid plate?

As long as you have the e brake on or wheels chocked, and the Trans or T case supported Id have no problem doing so.

the tranny mount and skip plate are ALL IN, one.
this is all you need to know.
the transmission must be supported.
the FSM covers this. a free real book for jeeps, YJ, need it ask, its free ,even googling carefully. I have it and can post it to PDF drop

you posted lacking the year. nor what transmission is there or engine (called the drive line)
parking brakes only work on rear wheels. and it is not and e-brake it is only parking brake, but im not nit picking, it is a fact. (just made mine work yesterday)
all new brakes and new cables. to them.
chock front wheels front and back if front is not lifted.
'
FSM words (terse)

TRANSMISSION REMOVAL AX5 , stick shift mine,
(1) Shift transmission into first or third gear. Then
raise vehicle on hoist.
(2) Support engine with adjustable jack stand. Po-
sition wood block between jack and oil pan to avoid
damaging pan.
(3) Disconnect necessary exhaust system components.
(4) Remove skid plate.<<<<<<<<<< bingo
(5) Disconnect rear cushion and bracket from
transmission (Fig. 5).
(6) Remove rear crossmember.

3 transmission and 2 engines, and details are not the same. it's the FSM how to work any tranny.
bolts a and b free the mount 4.0L

safe to idle in neutral ax5 and add, NV231 4wd lever in neutral too. double neutral, , if have slush pump A?T , park is safest.
if engine leaks exhaust before the cat the fumes can be full toxic with "CO" carbon monoxide the silent deadly killer.
CO makes the red blood cells can no longer transport OXYGEN. < off to ICU you go.
if fuel pressure is way too high (60psi) CO goes max!. never forget that on any car, MPI.

now a safer way
get rubber hose. 3foot long
one end to ear.
i duct tape (yes we love it) a 12" stick to the end, of hose. ( a $5 home made tool; max)

then use stick as tool to wave hose end ever every inch of exhaust system. the stick prevents burns off 2000f degr. pipes.
I use a fan in my shop to blow , fresh air to under car so not get poisoned.CO
engine at idle wheels chocked
I wave it and I hear load putt putts. bingo found it. (this is a basic basic skill set) cars.

my last found was SST exhaust manifold cracks 4.0L jeep, off it came and rewelded it, and found the header pipe mount cracked in half. allowing the whole
exhaust to wag like happy dogs tail and will repeat unless cured, that. too.

dont get burned, and no not sniff CO. (the latter is easy fatal)

folks that BBQ indoors get CO death. (charcoal burns nasty) same and RV with generator muffler leaks to cab ,easy.