Ok, thank you for all the information so far, I'll try to hammer out my understanding now...
1) ~なければならない、~なけりゃならない、~なきゃな らない (THE NUANCE OF NARANAI)
'Have to' regardless of the doer's will. The ~naranai is what gives it this effect. After all the various other ways to say 'have to' and their implications, though, I'm a little confused as for what situation this would be used in. Perhaps a teacher telling her students, 'You have to fill out your names here', お名前をここで書き込まなければなりません.
2) ~なければいけない、~なけりゃいけない、~なきゃいけない (THE NUANCE OF IKENAI)
Something the doer can choose not to do, but has to do. "Tanaka, you have to come to the party [or we'll miss you]!" 田中くん、パーティーに来なきゃいけない!
3) ~なければだめだ、~なけりゃだめだ、~なきゃだめだ (THE NUANCE OF DAME)
The doer MUST do something. 'To win the game, you must throw the ball through the hoop'. Is this correct?
4) ~なくてはならない、~なくてはいけない、~なくては だめだ (「~なくては」の代わりに「~なくちゃ」を 使えます)
I'm not sure about this one. What's the nuance implication of this construction?
5) ~ないといけない、~ないとだめだ
An advice-giving have to. 'If you want to get over your relationship, dear friend, you have to let him go'. That sort of have to? And I'm guessing the nuances of ikenai (you can choose not to do, but have to) and dame (you must) are still in play?
6) ~ねばならない
A contraction of ~nakereba naranai that is classical, so this 'have to' is only found written.
7) ~べきだ
A have to that is generally/morally correct. 'If you don't want to be hungry, you have to eat'.
8) ~ざるを得ない、~ないわけにはいかない
The doer doesn't want to, but has to for some reason. 'If you want to save your friends, you have to go down this dangerous mine shaft!' (Okay, getting a little uncreative with examples). ~naiwake ni wa ikanai is more colloquial.