"p into back and then front of next st (increase made)"
Does that mean I K into the front and then back? Or would it still be K into back and then front?
You can knit into the front and back. For purls it's just easier to do back/front for an increase.
"Row 1: *k6, (p1, k1 tbl) twice, p1; rep from * across."
Would I do the P1 into the back loop still?
I'd say go through the back loop as well.
"3/3 LC (3 over 3 left cross) Sl 3 sts to cable needle and hold in front of work, k3, then k3 from cable needle."
This is the cable part of the pattern so would it still be kept as K3 or would I have to convert it to P3?
The cable cross would be on the right side of the work either way, so keep them as knits.