The Wikipedia article details how Galicia became part of these other kingdoms but it leaves me confused on how a "kingdom" can be part of a greater kingdom. Normally I thought that with feudalism a kingdom would be downgraded to a duchy or county or something like that. Was the case of Galicia maybe a personal union thing?
This list is wrong in various points and I think that's why you are confused with the dates. Let us go one by one.
The kingdom of Galicia does not appear in the Peninsula until the early 10th Century, that is, when king Alfonso III of Asturias had his domains divided between his three sons: his firstborn, García, was to rule over the Duero valley from León (which is, incidentally, the origin of the kingdom of León); while his younger brothers, Fruela and Ordoño, were given Asturias and Galicia, respectively, to rule as kings, as long as the primacy of García over them was respected. You may understand this not as a division of the kingdom itself, but as some kind of viceroyalty made to ease the administration and avoid dynastic troubles.
Thus, the date that Wikipedia gives you for the creation of the kingdom, 409, is wrong. But not completely. During the last decades of existence of the Western Roman Empire, as you may know, certain gropus of germanic peoples crossed the borders and established themselves on imperial territories. The visigoths and Franks are, probably, the most known, but they were hardly the only ones: in Christmas 406, a large group of these peoples crossed the Rhine onto Gaul and, from there, they reached Hispania. Among them were the Suebi, who managed to establish themselves in the northwestern edge of the Peninsula, the roman province of Gallaecia -named after the Gallaeci tribe that inhabited the lands prior the roman conquest-, and form a small but active kingdom that lasted for some two centuries.
Our information on this kingdom of the Suebi, of Gallaecia as some call it, is close to non existant (there is, actually, a literal source silence of over 80 years between the first few and last kings). They only reappear late in the 6th Century, and only to be finally conquered by the Visigoths, with whom they must have had a decades-long struggle.
With its conquest, the suebian kingdom disappears, but not the suebi themselves, who left interesting remains in the region, such as a specific religious administration and typical suebian names not found in the visigoths, some of them even making it to the future asturian royal house, such as Vermudo, which comes from suebian Veremundus. This created a particular identity between the galician nobility on later centuries, when the region was under the suzerainty of León and Castile. And that bring us to the High Middle Ages.
The medieval kingdom of Galicia was only independent on very specific moments, usually during successions. The division of Alfonso III resulted in a civil war between the brothers, and between the sons of Fruela and Ordoño, until the last ones, specifically king Ramiro II, reunited the three territories under the same ruler. Galicia would remain united to León for a century, until 1065, when king Fernando I had the realm divided between, again, his three male sons: Castile for Sancho, the eldest, León for Alfonso, and Galicia for García, the youngest of the three. However, Galician independence was not meant to last and, not even a decade later, Alfonso VI had managed to replace both his brothers (Sancho had been murdered and García imprisoned) and rule his father's realm alone.
The next division would come in 1157, when the grandson of Alfonso VI, Alfonso VII, divided the Kingdom of León between his two sons, Fernando and Sancho. However, this division was different, for Sancho inherited Castile and Toledo and Fernando, León and Galicia. Since then, Galicia would remain as one with León (though independently in laws and identity, for they were not one, but two kingdoms), which eventually reunited with Castile -we'll get there later-, and later with Aragón to create modern Spain.
As the dates for this one are right, I'll go briefly over it: the visigoths ended in Spain after being pushed off of southern Gaul by the Franks of king Clovis in 507, though their kingdom had existed there for over a century already, and for another two hundred years they became lords of Hispania. They expelled the eastern romans from the Mediterranean coast and conquered the Suebian kingdom, but an endemic situation of crisis caused by weak kings, powerful nobles and civil wars made them an easy prey for the growing Caliphate, whose troops defeated and killed king Roderico in 711 and swept over the peninsula uncontested. Most of the Visigothic nobility retained its status as landlords by recognising muslim overlordship, but some fled north, to the mountains, where a resistance soon grew under Pelayo and slowly consolidated a small principality in Asturias.
Unlike Galicia and the Suebian Kingdom, which had no explicit political connection, León is the direct successor of the Kingdom of Asturias, as it is no more than a change of names: with the expansion southwards, the old capital of Oviedo, across the mountains, proved to be far from where the attention of the king was required, in the frontier, and so Alfonso III started expending more and more time in León, which eventually became the permanent seat of the kingship (although the court was itinerant and there was no fixed capital).
León became, in the mid 10th Century, the biggest player of the Península, but then entered a period of weakness due to dynastic conflict and the strengthening of the nobility, that even managed to depose kings, such as Sancho I. This situation was worsened by Almanzor's activity in the late 10th century, when León was put on its knees and its most important cities, Santiago and León itself, where sacked by the muslim commander.
Between the noble families that arose within León's weakness, one achieved a degree of almost parity with their kings, that of the Counts of Castile. Castile was, basically, the eastern third of the kingdom, a land of frontiers and fighting, and its lords, of which the most important were Fernán González and his son and grandson, achieved enough power to marry even with foreign royalty: eventually, the county passed to the king of Pamplona by virtue of inheritance, as the last male count was murdered without issue, and Castile was granted to the younger son of the king.
This son, Fernando, would go on and defeat the last king of León, Vermudo III, taking the crown by virtue of his wife, who was incidentally Vermudo's sister, and so Castile and León reunited under a new dynasty. This Fernando is the one I mentioned earlier, and he was the responsible for elevating Castile to kingdom level -for it was a county-. Castile would go on and exist along León for three generations and separated from it for another three, until 1230, when King Fernando III of Castile inherited the crown of León from his father, Alfonso IX, reuniting definitively both thrones. That 1230 that you mentioned in León is, then, the moment when the kingdom lost its independence. It continued to exist, however.
During the last two centuries of the Middle Ages, the situation in the Peninsula was pretty much the same: Portugal on the West, the Crown of Castile in the center, that of Aragón, in the East, and the muslim Nasrid Kingdom of Granada in the south.
The conquest of Granada and Navarra and the marriage of Isabel of Castile and Fernando of Aragón, the Catholic Kings, in the last decades of the 15th Century, allowed for the unification of most of the Peninsula, which was consolidated under the Habsburg dynasty that entered the Peninsula by the marriage of Juana, heiress to Castile and Aragón, and Philip the Fair.
This was, as you say, a personal union. Both Castile and Aragón remained independent in pretty much everything, their administration, Courts, tribunals and institutions being independent, the only in common between them being the king they shared. The kingdoms became, then, something like provinces of the Spanish Monarchy, and as such remained for centuries. The 1715 that you get for Castile represents the starting point of Bourbonic rule in Spain, and the abolition of Aragón's independent institutions and its integration to those of Castile.
However, Spain was, at that point, was no more than a collection of several kingdoms -administratively speaking- and it would remain so until 1833, when Fernando VII died and his widow María Cristina, acting regent of their daughter, Queen Isabel II, abolished the kingdom-based administrative system in favour of one of provinces, in an attempt to avoid a civil war which does not concern us now. So, all in all, the date of the dissolution of all the medieval kingdoms of Spain is actually the same, 1833.
About your last question, whenever I discuss this I like to remember what Robert Baratheon said to Ned Stark, "one king, seven kingdoms"; this sums up pretty well what Medieval Spain was like. A kingdom was personal patrimony of a king, and as such, he may own more than one -picture it as if you owned, say, three houses-. This doesn't mean that there is one above the others, or that those are secondary or less important: the kings of León usually styled themselves as kings in León, Castile and Galicia, and it's only for commodity's sake that we don't get to call them by their complete titles. For example, the son of Fernando III, king Alfonso X, is usually recorded calling himself "Alfonso, on God's Grace king of Castile, Toledo, León, Galicia, Sevilla, Córdoba, Murcia and Jaén", but we commonly know him as Alfonso X of Castile.
Right now I'm away from home, but if you have any follow up or wish to see any maps, which I know would be useful but I can't link right now, just let me know!