Issue with kickdown using BPE cable kit

The delay when shifting into gear is probably torque converter drain back. When it sets for a while the converter drains back and since fluid bypasses in park it doesn't start filling the converter until you shift out of park. From now on put it in neutral and let it idle for 30 seconds and I bet you won't have that issue.

As for slipping going into 3rd, a front band adjustment is probably a good idea but I doubt it'll help since the front band is for 2nd gear. I'd try adjusting the throttle pressure cable a little tighter. The throttle pressure regulates pressure to the clutch packs and servo's to "tell" it how firm to shift. The more throttle input, the higher the pressure so it doesn't slip when applying a lot of power and when it's adjusted too low it'll slip like you say. If you don't remedy it you WILL toast the front clutch pack and lose 3rd gear and reverse