No iPhone notifications when using Apple Watch

That's kind of the point of the watch; you get the notification on your wrist rather than the phone.
The only thing I feel on my phone is when I receive a call (but I chose this)

 

It would defeat the purpose of the watch but you could put it in airplane mode. Your messages would alert you on your phone since the watch would not be visible to the phone. I don't think that's what you want though.

 

If you get a notification on your watch and ignore it, the phone should ping a few seconds later.

The intent is, if you're using your watch, then it's where you're paying attention. It works the other way, too -- if you're using the phone, it will display a notification first, then the watch shows it later.

The "notification hierarchy" works the same way across all your iCloud devices, first appearing on whichever device you're using. When I'm at my computer, a text will pop up on iMessage on the computer before it appears on the phone or watch.