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.

It gets the benefits immediately

You get the Companion's Bond feature at 3rd level, at the same time as you gain the Animal Companion feature.

The Companion's Bond feature says:

In addition to the areas where it normally uses its proficiency bonus, an animal
companion also adds its proficiency bonus to its AC and to its damage rolls.

This applies from 3rd level and onwards, so it gets a +2 to its attack rolls, ability and skill checks, saves, AC and damage, and loses Multiattack if it had it.

Once you reach 4th level, and every level beyond that, it gets an extra hit die:

For each level you gain after 3rd, your animal
companion gains an additional hit die and increases its hit points accordingly.

This means that your giant weasel's stats look like this:

Giant Weasel

Medium beast, unaligned

Armor Class 15 (13 +2 proficiency)

Hit Points 9 (2d8)

Speed 40 ft.

Ability Scores
(+2 proficiency to all saves)
STR 11 (+0)
DEX 16 (+3)
CON 10 (+0)
INT 4 (-3)
WIS 12 (+1)
CHA 5 (-3)

Skills Perception +5, Stealth +7

Senses darkvision 60 ft., passive Perception 13

Languages —

Challenge 1/8 (25 XP)

Keen Hearing and Smell: The weasel has advantage on Wisdom (Perception) checks that rely on hearing or smell.

Actions

Bite: Melee Weapon Attack: +5 to hit (+3 +2 proficiency), reach 5 ft., one target. Hit: 7 (1d4 + 3 + 2 proficiency) piercing damage.