'So what exactly is 4 - 6 generations? [...] Are all of those considered generation 2?'
A generation of yeast is considered as having gone from pitching to post-fermentation collection. So if you ferment a batch with new yeast and collect five jars, those are all first generation. If you pitched every one of those jars into a new batch and collected five jars from each, you'd have 25 jars of second generation yeast. Continuing with the same method, you'd end up with 125 jars of third generation yeast, &c &c. Obviously you're not going to do this (there's no reason to collect more yeast than you need), but this is how generations work.
If you were to collect just two jars from the first batch, pitch only one new batch and collect two more jars from the second batch, you'd be able to choose between the two jars of second-generation yeast and one left-over jar of first-generation yeast.
'Is this what commercial breweries would do?'
Some do, though slanting is not a great way for large-scale commercial production. Typically isolated strains would be stored in liquid nitrogen and built up in a propagating flask and then in a larger propagator. It's not that slanting couldn't work, it just requires re-culturing for long term storage (usually twice or more a year, with a possibility for strain drift), while samples stored in liquid nitrogen can stay viable, and not experience genetic drift, for many years.
Most 'craft' breweries I've seen will get their yeast in pitchable quantities from a laboratory, often 1-10 liters. Much like malting, the most sensitive parts of yeast propagation have moved away from being done in the brewery and formed a specialized subset of people and companies.
'Could I just buy a microscope and learn how to do a yeast viability test on my own? Or do I need a microbiology degree for that?'
Yes - There are a tons of places on the internet that will teach you a great deal about yeast and how to handle it.
No - you don't need a degree for this any more than you need a degree to brew beer in the first place.
'Whats the ideal temperature to keep the yeast? [...] Will the yeast die if its to cold?'
As cold as possible without freezing. Yeast will be increasingly dormant at low temperatures, surviving on their own energy reserves. Lower temperatures slow the yeast's metabolism and therefore prolong storage life.
Freezing will form ice crystals which basically tear apart the yeast cell from within. Definitely not an ideal situation. For home storage, the fridge is fine. Slants should last 6 months to a year before you would consider re-culturing. Jars, handled properly, can stay good for months too, though you would typically want to use them to make a starter to pitch, rather than pitching directly.