1. Are Shaggoths good?
Yes and no. Large target without any real saves, a lot of points in an already elite army. Certainly fun, not so competitive.
2. Are Warshrines good?
I'd learn more toward the 'no' here. They CAN be good, but you have to seriously invest in them to make them so. Warshrines allow an extra roll on the Eye of the Gods table within a certain range (I think its 12") - which buffs Chaos Champions after they kill enemy champions - and the warriors called Chosen, a special choice, start the game with a buff. Yes one buff makes you a Daemon Prince, but another makes you a spawn. It's Chaos.
If you want to take a warshine, don't bother with warriors in your core. The Chosen aren't much better at all than a warrior stat wise, and preform the same role, albeit with a buff. Take Chariots/Dogs/Marauder horsemen, and then blocks of Chosen as your (killy) anvil to earn the Warshrines pay.
So you can already see how expensive it works out to be. The Warshrine is fun, and a cool idea, but its quite gimicky. You have to use your special choices to truly justify it, and your main core unit (chaos warriors) then has little to no purpose. Moreover, other than warriors, the core choices are nowhere near as good, barring chariots, and even then the special choice chariot for 10 points more is far better, in any shape or form. Marauders are awful, Marauder Horsemen can't compete with the special Fast Cav choice, dogs are simply Dogs, and again the Chaos Chariot loses out mostly to the Gorebeast Chariot.
On the good side, the Warshrine is hella strong. It will hardly ever die. Poor opponents may concentrate fire on it, but others who have played it before probably won't. It's just not worth the investment.
3. What base size is the Hell Cannon on?
As above
4. Are Hell Cannons good?
Ever single Chaos list 1000 points+ should have one. I take two whenever I can. It's simply excellent. It has a pretty bad misfire chart but when it hits, it hits, and it's a bloody monster in CC. No other war machines compare!
5. What would be cool legs for chosen? I'm planning on using the Blood Angel Sanguinian Guard torso's (with modeling adjustments) but I'm not sure which legs would look right underneath them.
Sounds cool!
6. Are Deamon Princes all that they're cracked up to be?
No. Cost more than a Chaos Lord and a Chaos Lord can preform any role better. Your DP can be a wizard, but then you're making some serious points investment in your large target. Best route points wise is a Sorcerer Lord, Level 4, and exalted heroes made for CC.
The main things to remember with Chaos are twofold: 1. It's a very, very elite army. You will almost always be outnumbered. Units must earn their points value back. 2. Players will know exactly what your game plan is: get into CC. Very good opponents will do everything they can to avoid it, whittle you down, and dominate magic.
It usually comes down to 'can my units get into CC while they're still effective?' if yes, Chaos mostly emerge the victor. This is why Sorcerer lords are taken so often, as the little shooting Chaos has, makes their magic far more potent, and their Sorcerer's can hold their own in combat.
A good Chaos player will have several units preforming several different functions (CC, Warmachine hunting, sniping etc) which, when used as a whole, is to do everything they can to get their units in CC without taking too much damage. We can't lose units like other races such as Skaven, Empire etc and carry on; every crippled unit is a huge blow to us.
Played Chaos for years so if you have any Q's feel free to PM me! My answers certainly aren't as fun but they're my honest experience.My best overall advice is build your army and see what you like the best!