Yes.
The soulknife ability clearly states that if a soulknife has Powerful build the knife it manifests is larger to match the ability. So you would deal increased damage due to having a larger knife than say.. a regular human.
By design this also means if you can find a way to get Enlarge Person to increase your size, then you would also benefit from the soulknife damage increase granted by powerful build as well.
Form Soul blade states:
A soulknife must choose the form of her mind blade at 1st level. She can either form it into a light weapon, a one-handed weapon, or a two-handed weapon. Once chosen, her mind blade stays in this form every time the soulknife forms her mind blade. The light weapon deals 1d6 points of damage, the one-handed weapon deals 1d8 points of damage, and the two-handed weapon deals 2d6 points of damage. All damages are based on a Medium-sized creature wielding Medium-sized weapons.
This constrains the Soul blade to a certain type of weapon, such as longsword, shortsword, trident, etc. Whichever you choose when you first manifest it. The size changes based on your own size, which may have variable scaling.
While Netzach makes a very good point, I'd like to add in the die dynamics.
Although the average is very similar the two situations behave very differently.
2d6
Rolling 2 dice creates a bell curve distribution of the possible values. In this case the average (7) has a chance of 1 in 6, while the max has a chance of 1 in 36. Half of your rolls will be 6,7, or 8. With the addition of more dice this curve gets steeper. 4d6 would have an average about 14 (1 in 36) and a max of 24 (1 in 648)
1d12
All numbers have an equal chance of occurring (a uniform distribution), thus your chance of a 12 is 1 in 12. Only one quarter of your rolls will be a 6,7, or 8. And on a crit, 24 is 1 in 144. You have a better chance of rolling higher than average, but also a higher chance of rolling lower than average. Many would argue that this balances out. But when you are in a battle and need a 12, your chances are better with the axe.
In summary
Some prefer the Greatsword because the high chance of average damage is "slow but steady".
Others accept the risk of a low roll with the Great Axe in order to have a better chance at high damage.
Best Answer
Question 1
What do they convert into?
The Weapons and Size section on page 220 of the Player's Handbook says that:
This is followed by tables indicating the damage die size progression for each of one- and two-handed weapons:
So, since a medium longbow (which is two-handed) does 1d10 damage, a large one does 1d12. This increases the average damage from 5.5 to 6.5. Similarly, since a medium greatbow (which is two-handed) does 1d12, a large one does 2d6. This increases the average damage from 6.5 to 7.
Question 2
If I can use weapons one size larger, can I use this to have two two-handed weapons in my original size?
Bugbears and Oversized are defined in the Monster Manual on page 276.
Bugbears are medium size. Oversized says that:
Since you can already use medium weapons as if they're medium, all this says is that you can use large weapons as if they were medium. It does not make you large, nor does it change the size of the large weapons, nor does it let you use weapons of any other size as if they were any other size, i.e. you can still only use small weapons as if they were small, not as if they were tiny.
Therefore, no, you cannot wield two medium two-handed weapons. You still treat them as if they were medium.
Question 3
If number 2 works, does it work with longbow/greatbow or any other ranged weapon?
We've already established that number 2 doesn't work.
In the previously mentioned Weapons and Size section, it also states that:
Although it's not written explicitly, the examples make it clear that this rule is intended to apply to all creatures, not just Large ones, i.e.:
In this case, it appears that you can wield a small two-handed ranged weapon, such as a small longbow, in each hand. This doesn't make much sense, and I would expect a good DM to rule against it on those grounds, but technically I think you could fire the longbows while wielding one in each hand.
That being said, you couldn't reload either of them.
On page 217 of the Player's Handbook, under Load:, it says that:
All ranged weapons that use ammunition have the Load property, including all bows and crossbows.