FCC assigns a Call Sign for me to use, and its extendable to my family?
On GMRS? Yes. FCC rule 95.1705 says:
(2) Any individual who holds an individual license may allow his or her immediate family members to operate his or her GMRS station or stations. Immediate family members are the licensee's spouse, children, grandchildren, stepchildren, parents, grandparents, stepparents, brothers, sisters, aunts, uncles, nieces, nephews, and in-laws.
with this being said if I'm out with my Family using my call sign.
Can I add a suffix to distinguish the user on my family making the call?
Example;
Me- WRJC211
My Son - WRJC211(A)
Son 2 - WRJC211(B)
Son 3 - WRJC211(C)
Mother- WRJC211(M)
Dad-WRJC211(D)
You can do that if you want. However, it's a mouthful and unnecessary. Rules say that you need to give your call sign at the end of a transmission or series of transmissions, or at least every 15 minutes.
You can just call them by their given name: "Hey Bob, this is Joe, are you near the radio? WRJC211"
But it's up to you. If you want family to use the radios and feel comfortable with them, make it as easy on them as you can.