Reading the "Får får får?" and the resonse "Nej, får får inte får, får får lamm" it makes more sense to me now- even if it is a bit awkward in english- I can see why it's something that you would learn early in Swedish. Not something you would use everyday... but at least if the two words popped up in the same sentence you won't be quite as confused. I enjoyed learning this one, thanks!
In this case I think that the verb "får" is the one that Swedes use when the mean "to have offspring". While English speaking parents are going to "have" a child, the Swedish speaking parents are going to "get" one, but both mean that it is growing in a womb and will be born later. So those sheep "get" their lambs (which will become sheep when they grow up) by giving birth to them.
It's an adjective, but förra and nästa aren't regular. Most of the time, you'll want to skip the article with time. I'm not really sure to what extent idiomatics trump rules here, since there isn't an exact rule to go by, but I'd wager the overwhelming majority of natives would always choose to skip the article in the case of sheep, which tend to be very countable.
Yeah, sorry, I meant in this case. Generally speaking, you skip the article if it denotes a noun that's in a series. Years are a great example of that - but pandemic vaccines are not. It's not a strict rule, though.
Note that out of the 660k results I got for "det förra" on Google, looking through the first 50-60 results the overwhelming majority appear not to be applicable to this construction. There are many results like vi såg det förra veckan, which would mean "we saw it last week".