Question

The description of Otherworldly Steed's Slam attack says:

Otherworldly Slam. Melee Attack Roll: Bonus equals your spell attack modifier, reach 5 ft. Hit: 1d8 + the spell's level of Radiant (Celestial), Psychic (Fey), or Necrotic (Fiend) damage.

But the Steed has a Strength attribute of 18 for a Strength modifier of +4. Does it also add its Strength modifier to the attack and damage rolls of its Slam?

Answer 1

No, the steed doesn't add its strength mod to attacks.

The steed's attack modifier is "your spell attack modifier" in the spell's description (via the steed's stat block). If the steed used its own strength modifier, that would be specified (eg., as "your spell attack modifier plus the steed's strength modifier").

The steed's strength mod would be used for strength saves, strength (athletics) checks, etc..

From a game-balance/theory-crafting perspective, this makes sense: if it used the strength mod, that would break the bounded accuracy that 5e strives to maintain by having an attack modifier +4 higher than normal for your character. A rough rule of thumb is that advantage is approximately worth +5 on most rolls, so such a steed would always attack with much of the benefit of advantage without actually needing it (and benefiting from it again in advantageous situations). Further, a straightforward reading of the spell indicates that it's already giving you an additional attack every round (at least, if you're not riding the steed), further supporting the conclusion that it doesn't add its strength mod to attacks.

Answer 2

No to both questions

The Otherworldly Steed's Strength modifier is not added to the attack and damage rolls of its Otherworldly Slam action.

According to the Monster Manual:

Attack Notation

The entry for a monster’s attack identifies whether the attack is a melee or a ranged attack and then provides the attack roll’s bonus, its reach or range, and what happens on a hit. [...]

For Otherworldly Slam:

Melee Attack Roll: Bonus equals your spell attack modifier, reach 5 ft.

And if the attack hits, the rule in the Monster Manual states:

Hit. Any damage dealt or other effects that occur as a result of an attack hitting a target are described after the “Hit:” notation.

In the case of this creature:

Hit: 1d8 plus the spell’s level of Radiant (Celestial), Psychic (Fey), or Necrotic (Fiend) damage.

Nowhere in the attack roll or “Hit:” notation is the creature's Strength modifier included.

Side note

This reading holds true across all creature and monster stat blocks, not only those in the Monster Manual, but also the ones found in the Player's Handbook, as mentioned in Appendix B: Creature Stat Blocks:

This appendix provides stat blocks for creatures mentioned elsewhere in the book, particularly in the class, equipment, and spell chapters. See the rules glossary for how to read a stat block, and see the Monster Manual for even more creatures.