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.