Yes. You can sneak attack with any ranged weapon.

Throwing by hand, no. A pebble is not a finesse or ranged weapon.

Hurling with a sling, yes. See Sneak Attack in the PHB:

Sneak Attack (PHB p.96, emphasis mine)

Beginning at 1st level, you know how to strike subtly and exploit a foe’s distraction. Once per turn, you can deal an extra 1d6 damage to one creature you hit with an attack if you have advantage on the attack roll. The attack must use a finesse or a ranged weapon. You don’t need advantage on the attack roll if another enemy of the target is within 5 feet of it, that enemy isn’t incapacitated, and you don’t have disadvantage on the attack roll. The amount of the extra damage increases as you gain levels in this class, as shown in the Sneak Attack column of the Rogue table.

Sneak Attack considers the weapon properties not the attack type. While not using the usual modifies, as per spell description, you are hurling a stone with a ranged weapon.

Since you normally can apply Sneak Attack to a sling shot, why should using an enhanced projectile be worse than that?

A rogue cannot access the Magic Stone cantrip inexpensively, so I'd consider this clever synergy.
Also the resulting damage isn't imbalanced.

The unofficial ruling by Jeremy Crawford is also a yes.

As DM, I'd allow it to work, given how Sneak Attack and magic stone are worded.

Divine Strike will not work with primal savagery

Divine Strike says:

... when you hit a creature with a weapon attack you can cause the attack to deal an extra 1d8 ...

So, to be able to apply Divine Strike, you must be making a weapon attack.

However, when you use primal savagery the spell tells you that, as part of the effect, you must:

Make a melee spell attack against one creature within 5 feet of you.

With this spell you are not magically creating weapons that you can attack with as you want. You are actually making the attack as part of the spell's effects and the spell's effects tell you that the attack is considered to be a melee spell attack.

Melee spell attacks and melee weapons attacks are distinct and separate attack categories and they are not interchangeable.

As Jeremy Crawford has said:

[E]very attack is either a weapon attack or a spell attack...This distinction is built into the core [design of the game].

Thus, Divine Strike will not work to grant additional damage with primal savagery.