Aside from the specific changes that the Animal Companion feature makes to the wolf, it retains its statistics from the Monster Manual, so it begins with 11 hit points.
Per the class feature as described in Unearthed Arcana: The Ranger, Revised:
For each level you gain after 3rd, your animal companion gains an additional hit die and increases its hit points accordingly.
This works like any hit die. The wolf's hit die is a d8. When you level up beyond 3rd level, roll 1d8 + 1 (CON) and add it to the wolf's current hit point total to determine the new hit point total, bringing it to (3d8 + 3).
Alternatively, take the average value. Per DMG p. 276, the average for a Medium size creature is 4.5 HP per die on a d8. The wolf's new HP value is 17 (3d8 + 3).
If you increase the wolf's Constitution score when you receive an Ability Score Improvement, be sure that you update the wolf's HP total to account for the new modifier.
The formula for HP is: $$n\text{d}X + (n \times\text{CON})$$
where \$n\$ is number of hit dice and \$X\$ is the hit die value.
I have included both the option to roll for the new hit die or use the monster average because it is up to your DM to decide the method. Traditionally, taking the average is the approach used for NPCs and monsters, including animal companions.
Yes
The relevant passage does not say that you use the proficiency bonus instead of some modifier. Instead, it specifies that you add it:
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.
The Ape's entry in the MM has a +3 to its damage rolls (probably from its strength, like you say), and the feature does not eliminate that bonus, meaning that it gains +3+(proficiency) to its damage rolls.
Best Answer
Thrown weapons (like a rock) are a special case of ranged attacks - if the weapon thrown is considered a melee weapon, you use the ability modifier you would have normally used in a melee attack, rather than automatically using dexterity - so, usually your strength modifier, unless fighting with a Finesse weapon. From the weapon properties in the basic rules:
It looks like the attack modifier for the ape's rock throw has been calculated as if it was a melee weapon being thrown, rather than an intrinsically ranged weapon, so the attack uses the strength modifier. Improving the ape's strength should benefit both attack modes.