[RPG] Can enlarge person affect an eidolon that ‘s large due to evolution surge

1. Any 2 claws. Nothing anywhere says different, just that it requires two claw hits.

2. Only once per round. The wording you quoted doesn't specify "for each two successful claw attacks...", just when it gets two. The additional claws just make it easier to proc (and do their own damage, of course). Notably, however, unlike normal rend, the wording for Unchained Eidolon's rend evolution seems to leave it open for getting one rend per round per target. I would probably rule against that in my own games, but it's certainly a valid interpretation of the wording.

3. It is not, as I answered here. Rend is applied as bonus damage to the second successful claw attack, not as a separate attack, so it doesn't gain the benefit of (or take the penalty for) per-attack effects. This includes Power Attack, flaming, or any Damage Resistance the target has, making rend a useful way to punch through DR.

4. No, again, because it's not a separate attack. That damage is already applied as part of the claw that's getting the bonus rend damage.

5. The wording on that ability seems to leave that option open as well. Even a "half-and-half" approach should work, where if you hit the target during your full attack and then hit it again before your next turn with an AoO or immediate action, the second claw should get the rend damage. That's still going to be a bit of a GM call, and will probably be ruled the same way as the "once per round per target" option from part 2, since they rely on the same interpretation of that wording.

P.S. re #5: Running through a creature's threatened area only provokes one AoO from that creature, regardless of the number of their threatened squares you run through. Just an aside, since the question implies you'd get multiple free shots in that situation.

The +4 to Strength in mythic enlarge person is a benefit from that spell, not a general rule. Size bonuses and penalties don't stack - so your Large character (specifically because you're a race-builder Large character, and so have built-in size bonuses) neither gains Strength nor loses Dexterity from being affected by enlarge person, you simply go up a size category (and get the normal effects of changing a size category - your equipment weighs more, you weigh more, you can carry more, your size modifier to attack rolls and CMB increases, your attacks roll larger dice). If you were subjected to mythic enlarge person, you'd gain +2 Str and -2 Dex for the duration, as the +4/-4 would apply over your natural +2/-2.

See also: This question about size modifiers to Strength and Dexterity

Special thanks to DM_Blake of the Paizo forums, because I found this quote much faster there, regarding stacking penalties:

Pathfinder Core Rules, Chapter 9, Magic, page 208 (depending on printing) wrote:

The Spell's Result

Bonus Types

Usually, a bonus has a type that indicates how the spell grants the bonus. The important aspect of bonus types is that two bonuses of the same type don't generally stack. With the exception of dodge bonuses, most circumstance bonuses, and racial bonuses, only the better bonus of a given type works (see Combining Magical Effects). The same principle applies to penalties — a character taking two or more penalties of the same type applies only the worst one, although most penalties have no type and thus always stack. Bonuses without a type always stack, unless they are from the same source.