Why do we have 12 tribes?
Having 12 tribes is a sign of a great nation.  After Abraham prays for Ishmael, G-d promises: Regarding Ishmael, I have heard you. Behold I have blessed you...and I have made him  very great. He will beget 12 princes and I will make him a great nation.
(Gen. 17, 20)

How are the 12 connected?
Only the 3 are called "Fathers".  Genesis Rabbah 68, 11: Jacob took 12 stones and said, "G-d has ordained that He is affixing 12 tribes." He said to himself, "Avraham didn't affix them. Isaac didn't affix them. I, if the twelve stones go together then I know that I will affix 12 tribes.  Tanhuma Vayehi 15: Rivka or Adam could have had 12 tribes.

There are 12 tribes for 12 lands of Israel.
For Hashem is bringing you to a good LAND, a LAND of brooks of water, of fountains and depths... a LAND of wheat and barley, and grapes and figs and pomegranates, a LAND of olive oil and honey. A LAND in which not in scarceness shall you eat in it bread... a LAND whose stones are iron..brass. And you shall eat and be satiated and bless G-d for the good LAND which the L-d has given you. Beware.................... So you should lengthen your days on the good LAND which ... a LAND flowing with milk and honey. For the land which you come to inherit is not like the land of Egypt... And the LAND which you go over to possess is a LAND of hills and valleys... A LAND which the L-d your G-d cares for...And it shall come to pass (the second paragraph of Shema) and you shall be wasted from the good LAND which the L-d gives you.  (parashat Ekev) Sifrei: RSBY:the fruits of this land are not the same as the fruits of this land The LandS of Israel were predestined even before the  Israelite arrival.

G-d showed him the whole land unto Dan, and the Naphtali and the land of Ephraim and Menashe... (Deut. 34, 1-2) How could these lands be called by their tribes before the tribes inherited their portions?

The significance of the number 12
There are 12 good forces and 12 bad ones. Yalkut Reuveni Hayey Sarah, p. 219. ibid. Vayishlach p. 284: There could have been 15 tribes.  Malbim 35, 22; 49, 4: ARI; Ephraim and Menashe were supposed to be of Bilhah, but when Reuven mixed up the beds, he prevented this. In the future, Levi will be a tribe and it will be 13 like the number of "echad" which is higher in the Merkavah.

Ben Yehoyadah on BB 122a that Israel will one day be 13 tribes: the letters of Avrhm, yzhk, and ya'kv is 13.

Moses blessed 11 tribes, excepting Shimon, in order to fight the 11 "kelipot." It is 12 different ways to encode G-d's name. It is 12 expressions of G-d as the 12 months are 12 experiences of G-d.  
R. Nachman of Breslav:  49 letters of tribes corresponds to the 49 gates of repentance.

How do the tribes relate to one another (ideally)?
Bemidbar Rabbah 13, 14: Each tribe brought the same thing as a gift but each had a different intention. Deut. Rabbah: 9, 9: Moshe wrote 13 Torahs and put one in the Aron and 12 to each tribe. Magen Avraham OH 68: ARI: Each gate in heaven corresponds to a tribe and a custom of prayer.  Every Shul has 12 windows for the 12 gates in heaven for the 12 tribes based on Hulin 92a: The Fathers are like the bark and the tribes are like leaves.

Why is Joseph out of the picture?
Shelah: He is like a Father, therefore he is not a tribe.