I'm going to support @BBlake on this one by pointing out that this is theoretically a problem for basically any type of elemental interacting with the normal universe.
- What happens if I summon an Ice Elemental into a sauna?
- What happens if I summon a Mud Elemental on top of a sand pit?
- What happens if I summon a Lightning Elemental into an armoury?
- What happens if I put a Magma Elemental anywhere?
In all cases, it looks like the Elemental is basically in control of its form and its interactions with the "normal" universe into which it is summoned. If this wasn't the case, the elementals would have all kinds of problems and would likely start dying or massively destroying things just by being summoned into the world.
Ideally, Sacred Flame kills it (roughly) twice as fast
A simple formula answers this, with X being the number of rounds needed to kill the Shadow.
1d10 averages 5.5, halved is 2.75
1d8 averages 4.5, doubled is 9
For Firebolt, you hit on 8-20 (chance for crit), so .65 x (5.5/2)(crit bonus)x(XFB) >= 16
(Need to hit AC 12; roll 1d20 +2 Int +2 proficiency)
(14 / 13 = crit factor; (x+1)/x for damage with the chance of crit thrown in)
For Sacred Flame, you hit on 13-20 (and no crit) .4 x (4.5 x 2)(XSF) >= 16
With a DC of 11 (you only have +1 due to your 12 Wisdom)
(The Shadow makes a dex saving throw versus the spell)
Solve for XFB: 16/(.65 x 2.75 x 1.077) = 8.31 rounds
Solve for XSF: 16/(.4 x 9) = 4.44 rounds
As a practical matter, being able to kill it in ~4 rounds rather than ~8 rounds means that your Sorcerer with that meager 16 HP has a far better chance to survive the fight before having no offensive capability.
Actions
Strength Drain. Melee Weapon Attack: +4 to hit, reach 5 ft., one creature.
Hit: 9 (2d6 + 2) necrotic damage.
Two hits and your Sorcerer is down.
This is an encounter were the Cleric's "turn undead" would be a handy skill to have.
This whole problem also presumes that you stay more than five feet away from the Shadow. In the case of firebolt, since that is a ranged spell attack rather than a spell that calls for a save, if the Shadow closes to melee range your firebolt attacks at disadvantage.
(SRD p. 103; spell attacks) Most spells that require attack rolls involve ranged attacks. Remember that you have disadvantage on a ranged attack roll if you are within 5 feet of a hostile creature that can see you and that isn’t incapacitated.
Best Answer
The golem ignites.
First, there's the rules perspective. As you pointed out, the Fire Elemental ignites creatures and flammable objects. The Stone Golem is a creature. Therefore it ignites.
Second there's the in-fiction perspective, which is what the player was challenging. The Stone Golem is a lump of rock(s) animated by magic, but the Fire Elemental is literally the essence of fire itself. If it wants to burn something, then by golly that thing is going to burn. The Fire Elemental is essentially putting a sort of clinging magical fire onto the golem.
Alternately it may be the case that it's the magic that animates the golem that's being burned. As vaxquis pointed out in a comment, even creatures that are completely immune to fire are ignited by the Fire Elemental's touch, they just don't take any damage from it. The fact that the Stone Golem doesn't even have resistance to fire, much less immunity, suggests that something about it burns just fine.