Ok, I'm on my PC, which is old, on wireless that sucks, with a card that sucks too, so let's hope my intrawebz can stay connected long enough for me to send this.
Preface: A lot of these questions would best be answered by the chemist here, but I'll try to do the best I can to explain things, as I have come to understand them, but I would advise for you to wait for verification as I am not the expert.
A powerhead is like previously stated, not necessary in your tank, but it is for mixing your brackish water. A powerhead is basically a circulation pump, it has an output and and input nozzle, an impeller, and it draws water in the input and shoots it out the output. Make sure the powerhead you buy doesn't have some funky looking output nozzle so you can fit the plastic tubing over it. You will put the water and Prime in the bucket, and the estimated amount of salt to get your salinity to 1.003. Until you know for sure how much of your salt mix is required to get the amount of water you are mixing to that SG, specific gravity, go on the low side of any suggestion. The saltyzoo link I gave you earlier will be a lot of help when figuring out your salt levels, and your return water salt levels. Add the powerhead and let it mix the salt a minimum of several hours, best if left overnight. This may seem excessive, but you want to make sure you don't add undissolved minerals into the tank as this can be very harmful to your puffer.
Here is the powerhead I use for mixing and returning water to the tank:
http://www.thatpetplace.com/pet/prod/253894/product.web Mine is a much older model, so I'm not sure if that flat deflector on the output nozzle comes off of the new one, but mine doesn't have that, it's just a tube. Any powerhead/pump should work as long as it doesn't have a goofy output that won't allow tubing to slide over it.
Ok, the clear plastic tubing is something you get from Lowes or Home Depot because you will get robbed if you buy it from anything aquarium related. I recommend just bringing the powerhead in so you can test to see if the tubing will fit over the output nozzle. Here is a link to an example of the tubing, it should be in the plumbing section or spools, it's apparently called PVC tubing:
http://www.homedepot.com/h_d1/N-5yc1v/R ... ogId=10053
Yes, you can add the water to your tank with a bucket, but it is a PITA, and I prefer not lifting stuff over and over again, and I really like the steady even stream of water not disturbing my aquariums and inhabitants as much as a galloosh of bucketed water would.
You know that plastic that is used for presentation folders? It's transparent or translucent, kind of rigid, but mostly pliable, You can use a few of those to cover your substrate so the water will deflect and not stir it up. If you do this right, and you have rinsed it well, you could possible not have cloudy water for a couple days, but you still may. You can use a large plate as well, but I prefer the plastic because I can cover more ground.
A water softener? This is a question I can't answer, RTR can help you best here. But I can ask a few questions that will helpfully aid him in answering you. I am assuming that you have a water softener because your water is very hard, is the softener something you can disconnect or bypass temporarily? What is the brand/type of water softener? This is a great thing to have asked because the KH of your water is actually important in regards to cycling and how wonderful of you to think to check your water source ahead of time. (As I didn't and had a lot of problems)
You can get these tests so you can test your water for these parameters, general hardness and carbonate hardness.
http://www.bigalsonline.com/Fish_Water- ... tc=default
RTR may ask for these numbers. It would be great if you could bypass the softener, but I don't know if you can. Then if you are on city water you can just call them up to get those numbers. Is this absolutely necessary? Maybe not, but because I had an issue with my tap water that was related to the KH, and you brought up the softener, it very well may be quite important. RTR will be able to help you more here.
Something I just thought of, can you call or email the place you are ordering from and ask them what SG the puffers are in currently? As far as cycling the water as fresh, again please wait for RTR to answer on this, because he did mention something about low brackish not having an effect on the cycle time. Being able to skip the slowly raising the SG by .02 a week would probably be great.
As far as the "water being cycled", it's not actually the water that gets cycled. Again I highly suggest that you read the articles on the nitrogen cycle and aquarium microbes, but I will give you a shortened explanation for now.
The aquarium inhabitants will excrete ammonia as their waste product into the water column continuously, mostly from their gills, some from their poo. Ammonia is toxic to them, so the quick build up of this can kill them. In comes the first bacteria group that will consume the ammonia and convert it to nitrite. Nitrite however is still toxic and the quick build up of that can kill them. In comes bacteria group 2 that will consume the nitrite and covert it to nitrate. These groups of beneficial bacteria cling to surfaces, and are not present in the water column in any significant amount. They will be on your filter media, substrate, plants, any surface really.
Nitrate is much less harmful and can be present in the aquarium in small numbers without hurting the fish. Since puffers are scaleless, even the nitrates will have to be a less that with some fish. I like to keep mine below 5ppm, but below 20ppm is the max recommendation. There are several ways that nitrate can be removed from the aquarium, and that depends on the variables of the set up, but for this example I'll keep it relevant to your set up. 50% weekly water changes are going to help greatly with this, and should be plenty enough if you stock your tank moderately to lightly. The less fish the safer they will be from poisoning themselves in their own waste. And because your tank is low brackish, there are some plants that you can plant, water sprite being one, and it's one of my favorite plants. Plants need nitrate to grow, so adding plants will help keep this number down. I am not sure if you want to deal with plants or not, but I will say water sprite grows like a weed so it's not a difficult plant at all, and I find it to be quite beautiful.
The articles in the library explain how to cycle a tank, as you will need to complete this before you order your puffer so he doesn't die on you. So as the tank cycles, ammonia will rise first, then nitrite, then as the nitrate starts to rise, the ammonia should start to lower, then the nitrite. A cycled tank will have ammonia, nitrite readings of 0, but have some reading of nitrate. There are exceptions to the nitrate, it can also be 0 in a cycled tank, (Heavily planted, generous water changes, enough liverock in a marine tank, etc), but this is not likely in an initial set up, and doesn't exactly apply to your prospective set up. This is where the testing comes in, as you will be testing these parameters during your cycle.