Welcome to the salty side

Will try and answer all your questions but if I miss anything just post again.
Testing the tank:
Just fill it with water and leave it for a few days and make sure the water stays in the tank
Equipment needed for a basic reef setup:
Filtration:
Live rock - You will need around 25-30kg of LR for this system.
Powerheads - You need at least 20 times the tank volume in water flow for soft corals. I would aim for 25-30 times the water volume. So you need at least 6625lph of flow. Personally I would recommend getting two or 3 powerheads rather then one big one as this gives better flow patterns and you wont get as many dead spots with no flow.
The ones you have listed wont be any good. You ideally need powerheads with a wide angle of flow from them (i.e. not jet ones with pipe outputs). Three of the ones listed below would work and are the cheapest option:
http

/cgi.ebay.co.uk/Sunsun-Power-Head-Wave-Maker-Cups-Holder-3000-Litres-/280400140740
Korilia and tunze are some of the best but are expensive. All your main filtration will be done with the LR and water flow.
Your equipment - You wont need the airpumps. The canister filter isnt really needed but can be used. Remove all the bio media from it and use it for putting in phosphate removing media. Put some mechnical media at the start (ie filter wool or foam) and make sure you clean this out regularly (ideally at least once a week).
Lighting - Depends on your budget. I'm not sure what the dimensions of the tank or which will have an effect on how much light you will need. However a few T5 tubes will work for most soft corals.
Skimmer - Would highly recommend a skimmer. As you have not got a sump I would recommend getting a hang on the back one as it will take up less space within the tank.
Misc bits - Test kits for ammonia, nitrate, nitrite, PH. Heater (prob a 300w for this size tank). Thermometer. nets, salt. Ideally an RO unit so you can make your own RO water. RO storage containers for topping up the tank. Refractometer for checking salt levels (easier to use then a hydrometer).
Setting up
You will need to fill the tank with RO water (dont fill it to the top as you will be adding rock in later), then put in the heater and powerheads (if you get 3 power. Add your salt gradually and check the salinity often.
Once the salt level is right leave for 24 hours to mix properly.
Now you can add your LR. LR should not be kept out of salt water for any length of time. You do not need any lighting over liverock to keep it live but if you can light it it will colour up better.
Now its just a waiting game. Test the water often for ammonia and nitrite. Once these have both gone down to zero do a partial water change (mix the new salt outside of the tank) and you are ready to start stocking.
Edit: just saw your bit on not having access to RO water. Tap water will have a lot of things in that arent good for a reef tank. Not sure on distilled water to be honest. Check your LFS, most will be able to offer RO water, best bet though is to pick up an RO unit and make your own.