Absolutely do the tall upper and lower ball joints- 0.5" on the bottom and 0.5" or 1" up top. If you'll be using stock lower control arms you may consider welding a plate on the bottom to box it in and add rigidity- they can flex. If you're yanking the wheels off the ground then a bumpsteer correction kit might be a good addition in the name of safety. Do the math and buy the correct springs at a reasonable price but cut NO corners when selecting shocks. They are incredibly important and have much more of an effect than most think. You should buy the best shocks you can afford since you're trying to pull double duty, and having adjustability is cool but single adjustable leaves a lot on the table. If you can swing it buy D.A. with dials so you don't have to remove any bolts just to make changes- it's a pain and means less time driving. Bilstein, Edelbrock, KYB, and Afco are all excellent choices with the Edelbrock Performance IAS being very well suited to your needs. They were discontinued years ago, around the time big E sold their suspension line to QA1, another good supplier. They can occasionally be found on eBay but they're old now so....
I've driven multiple vehicles with defective or missing sway bars and I'll warn you: if you're used to having them you're likely to get into trouble when the car doesn't respond as you expect. Those quick 'accident avoidance' moves barely move the car at all, you'll be dealing with severe under/oversteer at different times and speeds, and you're more likely to roll the car or at the least turn the rear into the front. If the car is set up for the strip then it'll drive like it, which works here in KS but I don't know the topography of your area.
If given the option I'd always run front and rear sway bars on the street, especially with the additional weight of a BB. How big to go kind of depends on the weight, the springs, your driving style, etc but as 403Olds said I've seen people get ran over so I'd always run it on the street. A big front sway bar is better if your coil springs are light for easy weight transfer. A rear sway bar also has a tremendous effect on the steering, as does a Wolfe Racecraft style ARB such as on a "back halved" car like bracketchev1221 stated.
I believe you'll also want every single brace that you can add to your frame, and that topic has been covered very well over the years so there's some good reading material. In addition to the fine parts that UMI offers, H R Parts 'n Stuff, TRZ, and Trick Chassis have some things that you may like.
In a nutshell: a street car and a strip car have entirely different suspension needs and finding the correct setup to satisfy yours will take a bit of trial and error. Don't let the errors be deadly ones.
