Is it normal my disc rubbing with no tightened QR and to stop when I tighten it? Or it should not rubbing both times...?(Because now when I tighten the QR the disc starts rubbing in one of the pads. It is not rubbing when QR is open) Also is it normal my front tyre to be closer to one of the legs of the fork(not centered) when wheel is all the way in the dropouts?
The QR system was designed before the advent of modern disc brakes, and as such there are a few peculiarities such as the one you are experiencing. You should only worry about rubbing after clamping the QR.
No fork or drop-out is perfectly set. When you put the wheel in the dropout with the QR open, there likely be a bit of slack in how it fits in the dropout. This goes away when you tighten the QR. Depending on how your caliper is set this may make rubbing disappear or if you are unlucky, appear, depending on the final position of the hub relative to the dropout after clamping the QR.
Note that it is also important to properly center the hub in the dropout before closing the QR, otherwise the hub can get positioned off center. As mentioned this can cause rubbing or worse if you set the caliper when the hub is not properly settled in the hub, you will likely experience rubbing the next time you open and close the QR.
These problems are inherent to combing the QR system with the tighter tolerances demanded by disc brakes. I suspect this was one of the drivers behind newer standards such as thru-axels, which should be more consistent when fixing the wheel to the dropout (although I can't confirm as I don't have practical experience with thru-axels yet).
The practical solution is to carefully center the hub before clamping the QR (e.g., by placing the bike on the ground then opening the QR, allowing the wheel and hub to center, then clamping the QR). if you get rubbing the you will likely need to reset the caliper position or back the pads off. The goal is to set the caliper against a stable wheel position that you can recreate the next time you open and close the QR.
Any sort of rubbing is undesirable. But, when the QR is open, things are likely to rub because the wheel can move around in the dropouts, so the only thing that needs attention is the rubbing when the QR is closed.
Your front tyre should be centred between the forks when the QR is closed, if it's not and your disc is rubbing you certainly need some adjustment to your bike.
Possible causes of your problem are:
Wheel is not firmly seated in the dropout - @rider_x tells you how to check this in his answer.
QR skewer is bent - if the skewer has been damaged, the wheel may be pulled off-centre when you tighten the QR clamp. Take the skewer out of the hub and check everything is straight. Pay attention to the springs when you remove the skewer and be sure to put them back with the narrow ends towards the wheel.
Wheel is bent or not true - take it to a bike shop for repair.
When building bikes, I always make sure the QR is on as tight as it should be when ridden before anchoring the caliper to the fork. The fork will always flex a little when you loosen the QR so rubbing discs is not uncommon with loose QR's. It should NOT be rubbing when the QR is closed and your front tire should be centered between the forks. The offset tire could be bad dishing or the wheel/wrong size spokes on one side. Cheaper bikes may have skewed forks though which could account for the misalignment.
Today I realigned the calipers of the brakes. Now they are rubbing when QR is open but everything is good when I tightend the QRs. I hope there will not be any problems in the future. :) Thank you all!