Is photon something that already exists before measurement? Do photons actually exist when electromagnetic waves are travelling, or are they thought to exist only because energy always quantizes when it's measured?
And if photons really do exist, how their existence at one of the slits in double-slit experiment is proven? Doesn't the measuring already change the system so much that the interference pattern disappears? (ie. enough energy is captured, so that there's not much interference and then quantization result in the end is different)
 A: 
Do photons actually exist when electromagnetic waves are travelling, or are they thought to exist only because energy always quantizes when it's measured?

Since we know that photons exist (and we are able now to excite individual electrons and cause them emit single photons), nothing - expect the fringes behind slits - prevents us to think about light as a stream of photons. Wherever in the stream you place a detection screen, you are able to detect photons. I refer to this answer from anna v.
But does the light stream from a thermic source has an oscillating intensity (wich is the characteristics of a wave)? Apparently no and in this case to talk about a EM wave is a little bit reckless. Each photon oscillates with its electric and magnetic field components, but all together didn’t form a macroscopic wave, they oscillate chaotic in relation to each over and they are not polarized. In the case of a radio wave, where electrons get accelerated all together synchronic, you are able to measure the frequency of this wave and it’s polarization.
But even a radio wave consists of individual photons. As a reference take the tiny receiving antenna of a Medium frequency radio wave. The emitting antenna has lengths from 25 to 250 m, but the receiving antenna has a fraction of the length only. The signal gets decoded thanks the varying stream of photons which reach the receiver. 

And if photons really do exist, how their existence at one of the slits in double-slit experiment is proven? Doesn't the measuring already change the system so much that the interference pattern disappears? (ie. enough energy is captured, so that there's not much interference and then quantization result in the end is different)

Thirst at all, fringes occur even behind sharp edges (and single slits too). You can handle narrow enough slits and double slits as the superposition of single edges. Thoas Young would be in need explanation because he explained fringes behind slits with the interference between the waves from the two slits. Beside this the fringes (the intensity distribution) on the observer screen is static, it doesn’t moves to the right and left. But water waves are doing this. Why nobody replays this to him? Because in Young’s time they had no gif-images?:

The amazing thing is that we not taking in account that the photons somehow are interacting with the edge they pass. That is natural for Young’s time, but today? We know that on the edges surface electrons are located. These electrons obeying both an electric and a magnetic field. The photons have both field components too. And we don’t care about their interaction. I’m sure that a young scientist, varying the field components of the surface electrons will see changing fringes and will be rewarded for this discovery.
A: The photon (the particle) and the electromagnetic wave are the same thing. This particle-wave duality doesn't exist (or can't be seen) at our scale, don't try to imagine it, you can't. The particle and the wave are just two different points of view of the same thing, one doesn't disappear while the other is created.
More exactly, there is a single photonic field, everywhere, at an energy level we call the void. Adding specific packets of energy (quanta) into it creates an oscillation we call a photon. You can think of this oscillation traveling through the field using wave-like rules (it spreads) and interacting using particle-like rules (it focuses).
