How did scientists deal with electronics' problems before Kirchhoff and Ohm's laws?

This is a bit like asking how Aztecs built cars without the wheel: they didn't.

There was a chain of invention by scientists in the early 1800s building off each others work. Prior to then there was only electrostatics: Benjamin Franklin rubbing insulators together and noting charged objects attract and repel. Leyden jars.

In 1800 Volta invented the battery or "pile". This allowed experiments with a constant source, rather than ephemeral electrostatic discharge. That led to Davy inventing the arc lamp, and Ohm in 1827 quantifying this electricity. Then Faraday's work on electromagnetism, allowing generators, dynamos and motors.

Engineers turning it into a "product" came later. Swan and Edison both invented the light bulb; Edison, Tesla and Westinghouse fought over distribution.

If alternative laws which would not be accepted nowadays were used before this, would this mean that the research done until the discovery of the laws was wrong? Did Kirchhoff and Ohm themselves rely on wrong theories to create 'the good one'?

There's a little discussion of Kirchoff and Ohm here.

Kirchhoff's laws followed from applying Ohm's law but the way in which he was able to generalise the results showed great mathematical skills. At this stage Kirchhoff was unaware that Ohm's analogy between the flow of heat and the flow of electricity, which formed the accepted understanding of electrical currents at that time, led to an incorrect understanding of electrical currents. Since no heat flowed in a body at a uniform temperature, it was believed that a static current could exist in a conductor. Kirchhoff's work would, a couple of years later, lead to him to realise this error and to give a correct understanding of how the theory of electric currents and electrostatics should be combined.

Which suggests that the answer was yes - people were building off incorrect theory to some extent. In the case of Ohm, he was building off Fourier's work on heat conduction. Electrical conduction is similar but not exactly the same.

There isn't anything on quite the scale that "phlogiston" was in chemistry - a controversial popular theory that ultimately turned out to be wrong.

Before tubes or semiconductors were used there was no electronics science, only electric science did exist.

When Georg Simon Ohm discovered the laws of electrical resistance, he needed voltage sources with constant voltage independent of load current. He tried galvanic elements first, but their internal resistance was too high and the voltage not constant. He used thermoelements instead and the constant temperatures of ice water and boiling water. For different voltages, he used serial connections of several thermoelements. It is very remarkable that Ohm was able to find the law using such low voltages. He constructed and built the necessary current meter by himself. The input resistance of a lot of current meters used today would be much too high for such measurements. Ohm had very remarkable experimental skills in doing his reserach with thermolements only, the mean error of his measurements was less than 1 %.

Kirchoff and Ohm were in the vanguard of the developing science. Their 'Laws' were the result of trying to systematise what they observed.

Once they could measure something, current by deflecting a compass needle with a coil of wire, voltage by counting how many batteries they had in series, resistance by using lengths of wire of a given cross section, they noticed that certain values were always in a constant ratio. The rest, as they say, is part of the history of science.