I'm definitely not an expert on knighthood and its variations across Europe, but if you mean did an elite aristocracy of warriors exist in Italy and Spain, then I can confirm that it did. The broad structure of medieval Spain wasn't too different from elsewhere in medieval Europe. There are a lot of specific differences, and it changes a lot depending on whether you're looking at 8th century Spain vs. 14th century Spain, but broadly speaking you could do worse than thinking of Iberia as like warmer France but with the multiple competing Kingdoms like in Britain. Again, that's a gross oversimplification, but you'd need an expert on feudal systems and/or knighthood to really explain the differences and their significance.

Italy is a lot more complicated, and not at all my area of strength. Italy is a huge mix of different influences, including Byzantium, old Roman cities, Muslims (esp. in Sicily), the German emperors, and obviously the papacy to name but a few. Generally speaking, cities were more powerful in Italy than elsewhere in Europe, but landed nobility still existed and wielded significant power. Knights existed, but it's hard to map Italian politics onto anywhere else in Europe. There's also substantial differences between Norther and Southern Italy, and huge differences across different periods.

I mean you would think Rome would have sent some troops during the crusades as well.

On the role these two regions played in the crusades I am actually well situated to provide some context.

Spain was largely not involved in the traditional crusades, i.e. campaigns in the Holy Land between 1095 and 1291. They were explicitly excluded from the earliest calls for crusade, and while this pseudo-ban largely faded away, participation by Spanish nobles was generally discouraged. This was for a fairly practical reason, in 1095 Muslim rulers still controlled most of Iberia and Spain was considered to be on the front line of Christendom. The church had no desire to deplete one front to fight on another. Eventually, theological benefits (remission of sins and that sort) was extended to individuals who fought in the Iberian campaigns. The 1147 conquest of Lisbon was done in conjunction with individuals sailing from Northern Europe to fight in the Second Crusade, which tied the two campaigns together to some extent (and also represents that crusades greatest success). This is not to say that no one from Iberia ever fought in the Holy Land, but it was not a common practice. Iberia was host to numerous religious military orders, particularly the Hospitallers, and played a major role in supporting those groups with both funds and soldiers.

Italy was a frequent participant in the crusades, particularly the formerly Norman rulers of Southern Italy. The First Crusade occurred during a papal schism. The pope who called the crusade, Urban II, eventually won out over his rival, but at the time of his calling the crusade he was in exile in France since the German emperor, who supported the rival pope, controlled most of Northern Italy and ostensibly Rome. The Norman princes of Southern Italy, however, were independent of German rule and took up the First Crusade with gusto. Bohemond, one of these nobles, was one of the major leaders of the crusade and the Normans generally represented a powerful faction during the campaign. It’s worth noting that while Bohemond is labelled as a Norman due to his heritage, he was born and grew up in Southern Italy as did many of his companions. Bohemond’s father, Robert Guiscard, had been among the Norman mercenaries who relocated to the Italian peninsula (for more on that you’ll need a specialist in Norman history or Italian history, it’s a complicated story I’m not sure I’m adequately capable of telling).

Italian involvement in the crusading movement didn’t end there. Besides the obvious fact that the Papacy was in Rome and played a massive role in organising and perpetuating crusading culture, many Italian nobles contributed to the movement. Roger II King of Sicily never went on crusade per se, but his kingdom formed a major trading partner with the Crusader States established in the wake of the First Crusade and was a prominent player in the politics of the Holy Land.

Venice and Genoa were also major trading partners, and periodically provided fleets to support campaigns and sieges in the Holy Land. For example, Richard I’s siege of Jaffa during the Third Crusade was likely only possible due to support from Genoese and Pisan forces, including naval support. Venice was also responsible for the fleet provided to the Fourth Crusade, and eventually largely responsible for diverting that crusade first to Zadar, and then arguably were influential in the eventual taking of Constantinople (although there’s a lot of blame to go around for this one, everyone can get their share).

Essentially, yes Italy and Spain were major players in the broader crusading movement, even if their contributions often weren’t in the form of a king or major noble participating in one of the big numbered crusades. It’s worth remembering that the practice of giving the crusades numbers, and distinguishing between them as a result, is a product of modern historical writing. At the time they would not have been treated that way, and in fact many people participated in behaviour we would recognise as crusading outside of the big numbered campaigns.

Source:

Thomas Asbridge’s The Crusades is probably still my favourite accessible work on the subject.

Jonathan Riley-Smith’s The Crusades: A History is also an excellent book, and few people have had a greater influence on the field of crusader studies than Riley-Smith.