All right, bro - I went through the exact same stuff you are describing -- wouldn't believe groups did not work - apparently, people don't use this feature much. I used this feature completely on my previous phone and would be ready to return the damn phone if they didn't support groups correctly.
You mentioned two issues in your original post:
1.
Issues with contacts being added to google group and not phone
This is controlled by your contact settings. Go to Contacts ->Menu -> More -> Settings to get into Contacts Settings. In here, enable the accounts you want to enable when creating new contacts. In my case, I have only enabled Phone Contacts. By default, it enables all applicable accounts - so in my case, it was Phone, Gmail, Corp Email - and I was seeing personal contacts in my work outlook. After updating this setting to be only Phone Contacts, you should be able to resolve contacts going into google issue.
2.
Unable to associate contacts to contact groups
This took me a while and I finally figured it out - very annoying that there's no proper documentation as to how to do this. This is what you need to do:
a) create New Contact Group (Contacts -> Menu -> Display Groups -> Create new group)
Here comes the weird part - you can NOT directly create contacts in the group. Although it gives you the add contact option under menu, it will add the contact under all contacts. What you need to do is edit the contact group and pick which contacts you want in the group.
b) select your contact group (will show no contacts initially)
c) Menu -> Edit Group
d) Click Add. Now it lists all your contacts and you can check the ones you want under your contact group. Select the ones you want and click Done.
e) Click Save.
Tada
Hope this helps!