Good question! So the main issue here is really one of chronology. The most basic answer is that the Kingdom of Alba comes after Pictland or Pictavia as a political kingdom. But that alone doesn't really explain everything so I'll try to lay it out as clearly as possible before moving on to your final question about Pictish society and culture.
The Latin word Picti started out as a pretty generic term for all of the "barbarians" who lived north of Hadrian's Wall. There are some Roman sources about them, but they're all heavily steeped in Roman stereotypes about barbarians, so it can be hard to glean meaningful information from them. The main thing is that Picti referred to their process of tattooing themselves, as it means 'painted' in Latin, something u/Libertat goes into more detail about in this answer.
(Picti may also be a pun on the Picts' own name for themselves - the Gaelic name for them is Cruithni which can mean 'painted' but can also mean 'the created (ones)', which is a pretty common name meaning for a group to apply to themselves. Hard 'c' sounds in Gaelic are 'p' sounds in the P-Celtic languages, so this would have been something like *Pritani in Pictish, which isn't all that far off from Picti and is actually related to the etymology of the word Britain itself.)
So to the Romans, Picti are just tribes of Celtic-speakers who were ethnically and linguistically related to the rest of the British Celts, with the only difference being that they lived in the part of Northern Britain that the Romans hadn't conquered yet. Some of the groups nearest the Roman border in places like Fife did have trade with the Romans, and some of them may even have been client-kingdoms. However, they were never fully incorporated into the Empire. They were definitely influenced by their contact with the Romans though, most notably in their adoption of monumental stone sculpture, something I'll return to later when discussing their culture.
Complicating matters here is that at the time, the word Scotti was used to refer to the Irish. In Latin, this covered everyone in Ireland as well as the people speaking Old Irish who lived on the west coast of Scotland. The thing you have to remember about Scotland is that the Highland fault line divides the country into an eastern and western half separated by mountains. Before the advent of rail travel, it was much easier to get from western Scotland to Ireland by boat than it was to get from western Scotland to eastern Scotland by land. So culturally and linguistically, Ireland and western Scotland were very connected in antiquity and in the early middle ages. The Highland fault line is thought by some historians to have isolated them from the linguistic developments that affected the rest of the Celtic languages in Britain and the Continent. This is why you ended up with a situation where western Scotland was speaking a Q-celtic language known as Old Irish whereas the rest of Britain, including eastern Scotland, was speaking a language on a continuum of P-Celtic languages.
So after the Romans leave, you have got the Scotti in the west of Scotland and the Picti in the east. However, as far as unified kingdoms went, there were initially a lot more than just these two. In the 7th and 8th centuries, the northern Pictish kingdom known as Fortriu gradually expanded across eastern Scotland, conquering smaller British kingdoms. This eventually consolidated most of Scotland into Pictavia, a kingdom ruled from the northern heartland in Fortriu but with royal centres dotted around its conquered territories.
Eventually, the Picts conquered Dál Riata too, and the union of these two is what led to the creation of the Kingdom of Alba. Here's where we get to the part about 'Scots' replacing 'Picts' as the main ethnonym of this kingdom. Although the Picts were the ones doing the conquering, it was the Gaelic language of Dál Riata that ended up becoming the main language spoken across the kingdom of Alba. Scholars are still not entirely sure why this happened. Old Irish was already a prestige language in Pictavia before the conquest of Dál Riata - for example, we have praise poetry in Old Irish surviving for the Pictish king Onuist son of Uurguist, who reigned from 732 to 761. Irish-speaking clergy, many from Iona, are thought to have been hugely influential in the organization of Pictavia as they converted to Christianity. Ultimately we don't know why exactly the Pictish language disappeared, but as the kingdom came to be dominated by the Irish language, so too did its people eventually come to be known as Scots.
The question of whether the Picts were "native" to Scotland is difficult to answer. The Neolithic people of Britain did not speak Celtic languages. I don't think there is a single scholarly consensus on how much population replacement occurred when Celtic-speakers from the Continent migrated to Britain. If there was a lot of population replacement, then the Picts, as Celts, weren't really "native"; but there may have been intermarriage and relatively peaceful cultural blending, perhaps the language being spread by a minority of elites as happened with the Norman influences on English in the 11th century, so it's really hard to say. Compared to the Romans and the later English, the Picts were "native", but compared to the Neolithic people, they might not have been. They are just as native as the Welsh are to Britain - no more, no less.
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I have always been a bit unclear about the Pictish involvement in the north west Highlands and who lived in that area generally during this period.
Dal Riada doesn't seem to have gotten much further north than than Skye and Picts are generally thought of as living on the east coast (as I understand it) so who was living in the north west Highlands? I know that there was some Picts (or at least pictish influence) in the area, I am from Gairloch and a pictish stone was found there, but I understand that this was a rarity.