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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 monster stat blocks.

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.

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 monster stat blocks.

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 monster stat blocks.

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

This reading holds true across all monster stat blocks.

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 monster stat blocks.