Am I stuck with the +1 dmg mod? Even though I have 16 str and thus get a +3 to dmg?
I'll start with this part of your question since the answer is a simple yes, you are stuck with +1 damage. You do however get two damage dice one level before striking runes become easily available (and you don't need to spend gold unlike the fighter who will probably upgrade his weapon at level 4).
Do I also get the +2 from wildshape making my attack roll = 10 ? And thus I can use that instead?
I do believe this is the intend of Wild Shape, yes. However, due to wording you couldn't get this bonus if your GM is being stubborn. Animal form says
If your unarmed attack bonus is higher, you can use it instead.
and Wild shape says
When you choose to use your own attack modifier while polymorphed instead of the form's default attack modifier, you gain a +2 status bonus to your attack rolls.
So going strictly by RAW, you could only chose to use your own attack bonus if it's already higher than that of the spell (which it isn't in your case with +8 vs. the spell's +9). And only if you choose to use your own attack bonus you get the +2 from Wild Shape. But since you can't choose it, you don't get the bonus.
Just to be clear, I'd totally give you the +2 regardless beause that's most likely what the intention was. But going by RAW, you could only get the +2 if your unarmed attack bonus was higher than the spell's without the extra +2.
If the character is wearing +1 handwraps of striking and turned into a Cat, would that mean they would get +10 Atk and 4d6 damage ?
No on damage.
Your Strikes do the damage noted in the spell.So Striking Runes, no. Strength bonus, no. And so on.
The spell's listed values supersede those.
There's some debate on the attack bonus, but the consensus appears to be you can add it to your PC's attack to see if it's better than the Wild Shape's attack when determining which to use. But the +1 never adds on to the spell's listed attack. The spell supersedes that too.
There's also some argument about "extra" damage, like that from Rage, Flaming Runes, and various other sources. As of a few days ago, there was no consensus and no word from Paizo.