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Let's say we are doing the double slit experiment with electrons. We get an interference pattern, and if we put detectors at slits, then we get two piles pattern because we measure electrons' positions when going through slits. But an electron interacts with other particles in a lot of different ways, e.g. electric field, gravity. Seems like the whole universe is receiving information about the electron's position. Why is it not the case and the electron goes through slits "unmeasured"?

Bonus question: in real experiments do we face the problem of not "shielding" particles from "measurement" good enough and thus getting a mix of both patterns on the screen?

There are time-scales related to interactions, or, equivalently, interaction rates. An experiment that measures electron interference needs to make sure that the time-of-flight of the electrons from the electron source to the observation screen is much shorter than any of the time-scales of possible interactions.

In interference experiments, we therefore define a coherence time for the interfering particles.

In real experiments, we do indeed face the problem of shielding particles from being measured by the environment, before they interfere. For example, in electron interferometers realized in solid-state devices, we have to go to very low temperatures, where the interactions between electrons and phonons become very 'slow' (their rate becomes very small). We also have to make sure that the devices are small enough that the Coulomb-interaction between electrons, which persists even at the lowest temperatures, does not spoil the interference (the decoherence rate due to electron-electron interaction does also depend on temperature: the rate becomes smaller with decreasing temperature).

As long as these interactions are weak and do not distinguish between the two slits, they can be disregarded.

Distinguishing which slit is which is the factor that causes the wavelike interference pattern to disappear. Experiments show that the more the path can be determined the more they look like single photons.

Here's some notes on a course where this is worked out explicitly for a Mach-Zender quantum interference experiment, where this continuum between "classical" and "quantum" is made mathematically explicit.

So yes, the more the experiment's electrons interacts with the "universe" in a way that the "universe" can gain information about which slit it went through, the more the "quantum interference pattern" disappears. This is a good intuition for why things at a macroscopic level behave classically: because the individual quantum pieces are interacting with the environment so much that all of this "quantum perserving" information leaks out.