Yes

Your companion uses your proficiency bonus rather than its own.

It is important to note that it says, without condition, that the companion uses the Ranger's proficiency bonus. This means that, from this point on, it's proficiency bonus is the Ranger's proficiency bonus. Think of it as erasing the value on the creature sheet, replacing it with the Ranger's value, and then recalculating everything from there.

Upon looking at the UA document again, this is further supported as such:

Keeping Track of Proficiency

When you gain your animal companion at 3rd level, its proficiency bonus matches yours at +2. As you gain levels and increase your proficiency bonus, remember that your companion’s proficiency bonus improves as well, and is applied to the following areas: Armor Class, skills, saving throws, attack bonus, and damage rolls.

While the wolf has a listed proficiency bonus of +2, nowhere is it stated that this bonus is already included in the damage of the wolf's attack. We may infer that this damage is from the wolf's DEX mod, as is supported by the fact that the base to hit is +4, which would be +2 from proficiency and +2 from DEX, matching expectations for standard application of proficiency bonuses. The wolf's damage should be increased by a flat 3 points, making the roll 2d4+5, the same as you have stated. Furthermore, applying +3 to the base 13 also matches your stated 16 AC, and replacing the +2 with proficiency with +3 also results in a +5 to hit when added to the DEX mod. It is not explicitly stated that the bite is a DEX-based attack, but this is the only value that makes sense from the creature stats. Your math looks right.

Bonus Answer: No, the damage dice of the wolf's bite attack do not scale. It is important to consider that the wolf companion is not intended to be as powerful as an extra PC, but is instead part of the power of the Ranger to whom it belongs. Applying proficiency bonus to a damage source is a benefit other classes do not have, and this comes from an attack that does not require the use of any of the Ranger's actions. Also, consider that this attack has the potential to knock a target prone, leaving it more vulnerable to melee attacks until it has the opportunity to stand. Such a tactical advantage is certainly not weak.

No

Tentacle Slam is not technically an attack.

In the stat block, the other actions include the phrase Melee Weapon Attack while Tentacle Slam has no similar verbiage. Tentacle Slam does not call for making an attack roll with a d20.

Tentacle Slam is an action that causes grappled creatures to make saving throws. It does not qualify as an attack for the purposes of the Otyugh's Multiattack ability.

When an attack is not an attack

This prior answer explains what counts as an attack in the game mechanics. That answer draws upon the passage from the Player's Handbook:

If there's ever any question whether something you're doing counts as an attack, the rule is simple: if you're making an attack roll, you're making an attack (PH, p. 194)

There are many actions that would likely be considered "attacks" in-universe, but which are not attacks mechanically. Examples include grappling and spells with save versus damage effects.

When reading a term in the rules, interpret it with the definitions in the rules. When deciding how NPCs/monsters react to something, use common sense.