I have a problem with an overtightened water pipe connector at the back of the toilet. It isn't an ordinarynut type connector, it has wings sticking out. Not enough room to get a grip with pliers

Do you mean the water drains too slowly causing the pan to fill up or the cystern won't fill fast enough from the mains pipe?
The sewage or waste pipe is not made of metal, not affected by rust/corrosion and should be pottery or plastic.
If slow to drain, there could be a blockage further down the outlet pipe to the main sewer, causing it to back up. If that is so, your plumbers are not professional, in my opinion.

if you have tried the one that comes with the toilet and this does not work...go to lowes and get a braided toilet supply that has rubber seats in both sides...these work great...good luck...

Install wax ring around base waste pipe. Make sure that 2 bolts are sticking out of a track under the toilet.Put toilet bolts through toilet holes and tighten nut with washer on each bolt to lock down the toilet. Now attache the water line that goes into toilet and bidet.

Mechanism that releases and shuts off water should be checked if any part has disconnect or improper connection.

Is it always low? Could be a partial blockage in the out flow pipe, causing the pump to stay on, or a sticking/dirty pump switch.

turn water supply off to toilet and disconnect water line from bottom of toilet tank. flush toilet to drain tank, use towel to soak up water inside that didnt flush.remove the 2 bolts ( 1 on each side of toilet), lift toilet up and move to one side and lay on it side, scrape off old wax ring from bottom of toilet, scrape off excess wax from pipe on floor. install new wax ring over drain pipe. stand toilet upright and lift straight up over drain hole and reset slowly to make sure each bolt goes through the holes on toilet, replace nuts and washers on toilet and tighten. hook water supply liune back up and turn on water.

The wax ring has gone bad. If the toilet moves when you set on it you might try snugging down the bowl bolts on both sides but be very careful not to go to tight you will brake the bowl.
If you decide to try to fix this your self with you buy a new wax ring buy the one with the horn in it they seal better.

Condensate can build up around the outside of a toilet bowl and that's often referred to as "sweating".
Condensate is caused when the air is very humid and an object (in this case, the toilet tank) is colder than the air.
Look up something called "dew point"; that describes the point at which the water that is suspended in the air will start to condense on cooler objects.
Dehumidifiers and refrigeration type air conditioners can help reduce condensate.
You can remove the float and stem by doing the following:
1. turn the water off and flush the toilet . Hold the handle for a while to make certain that you get as much water out as possible, then remove the rest of the water using a towel or whatever means available.
2. Look where the water supply line feeds into the tank. There should be a nut that holds the pipe to the tank. Remove the nut then carefully move the pipe a little out of the way... not much.
3. There should be ANOTHER nut that attaches the float assembly. Remove it and note where the rubber washers are. You may even want to replace the washers when you're done, to make certain that the tank doesn't leak.
4. When you remove that nut, the float assembly may stick a little, but gently work it "back and forth" a little and you should be successful.
Good luck

for the upstairs if the valve is fully on there is not much you can do other than replace water service to the house or heaven forbid repipe.Valves usually don't restrict fill volume.But you can take the supply loose from the tank and stick it in a bucket turn the valve on and check the flow.this should narrow it down.Down stairs repair or replace flush valve

The Gerber Ultra Flush Back Outlet toilet, that uses 1.1 Gallons of water to flush?

Just like any common toilet. Install the tank to the bowl of the toilet, set the toilet, install the flush mechanism in the tank, attach the supply line.

1.MAKE SURE you put the two plastic U-shaped spacers on the back of the bowl!
2.Install the round rubber gasket in-between the tank and the bowl.
3.Set the tank down on the bowl. Make sure you have the round gaskets that go up under the bolt heads, then insert the two bolts into the tank, and through the bowl. NOT installing these will cause a leak!

Tighten these bolts down, until the nuts are JUST touching the tank. (There is a flat washer that goes under each nut)
DO NOT tighten one bolt all the way, then tighten the other bolt!!
These bolts MUST be tightened evenly. Tighten one turn on one bolt, then go to the other bolt.
DO NOT overtighten, or you will break the bowl, and/or the tank!

Make sure your toilet flange on the sewer pipe is in good repair, and clean, from the old beeswax ring. It can be messy, even using gloves. Use a flat painters stick to help remove the old wax ring. Scrape off into a old can. The wax thin residue on the floor and flange is fine.
Set the new wax ring in place, remove the plastic liner that comes with the ring. DO NOT leave that thin plastic liner! It's just to install that wax ring!

Install the bolts that hold the bowl to the floor, or reuse the old bolts if they are in good repair.
Tighten EVENLY, and don't overtighten! You can crack the bowl.

Now install the toilet mechanism in the tank.
Here is a 1 Page, Tank to Bowl Installation Guide from Gerber. It is a PDF file. Let it download ALL the way before looking at it.
http://gerberonline.com/fileLib/Toilets%20Ultra%20Flush%20IS.pdf

Should this direct link not work for you, here is the page it came from,
http://www.gerberonline.com/Catalog.aspx?categoryID=2204&itemID=3514

It's on the right side -> Installation Instruction (PDF)