No, they are not related.
Bear has a very solid, established etymology going back from English to Proto-Germanic to Proto-Indo-European. The PIE root is *bʰer-, and it means ‘bear, carry’ all the way back to PIE.
Govern is a loan word in English that ultimately goes back to Ancient Greek κυβερνάω (kybernáō) ‘steer, drive, rule, govern’. The origins of this word are less secure – it may be a substrate word borrowed in from an language which was spoken in the area when the Greeks first arrived there, but which died out early on. Some inherited, Indo-European origins have also been suggested, primarily going back to a root like *kʷerb- (an unclear variant of the more well-established root *kʷerp- ‘turn’, related to whirl in English), but it’s rather a roundabout connection and not commonly accepted.
Certainly, if κυβερνάω is a substrate word, we can say for certain that there’s no etymological connection. But even if it is an inherited word, there’s no possible way to connect it to the bʰer- root. The crucial reason for that is that PIE *bʰ became pʰ in Greek (written φ, Romanised ph or f). Even if κυ- could be shown to be some sort of prefix, if the word had come from the *bʰer- root, it should have been κυφερνάω in Greek, which it isn’t.
As for your second question, that is more or less the basics of how linguists approach etymology, yes, but at a very primitive stage. It’s essentially a kind of ‘first step’ to identify words that look similar and have compatible meanings.
But after that stage comes the even more important part of checking sound laws. Over the past few centuries, historical linguists have accumulated an enormous body of rules that describe how particular sounds (and sequences of sounds) developed from one stage of a language to another. Given the number of languages in existence and how many stages of those languages are known, you can imagine the sheer volume of sound laws that have been identified – as well as ones that haven’t yet been identified.
The thing you always need to keep in mind is that it often doesn’t matter much how similar two words are now, because they may have come from the same source, but developed in separate directions over several thousand years to the point that they are now completely unrecognisable. For example, it’s fairly obvious that Icelandic tvo is cognate with English two; it’s less obvious, but still visible that Latin duo and Sanskrit dvá- are also cognates; but you’d have to be uncommonly astute to recognise without aid that Armenian երկու /jeɾˈku/ is also cognate.
In order to ascertain whether two words are actually cognates, you need to identify a likely source in a common ancestor of the two languages, and then go through all known sound changes from that common ancestor into each of the two languages to check whether the sounds would change in the right way to yield the words you started out with.