The Italian Wars were Italian primarily in the sense that Italy was the theatre and the prize fought over. The main contestants were the House of Habsburg and the House of Valois, and the main prize was the Kingdom of Naples. Actual Italian states played a secondary role, engaging in their usual games of thrones in the shadow of the clash between these two royal dynasties.
While pretty much all European states started raising similar kinds of pike and shot armies at some point during the Renaissance, Landsknechts refers specifically to the German kind, wearing the extravagant puff and slash style we know from the Warhammer Empire. They and their Swiss equivalent were indeed mortal enemies, with the Landsknechts having originally been chartered by Holy Roman Emperor Maximilian von Habsburg as a specific counter to Swiss pike, in an (unsuccessful) attempt to put an end to Swiss independence. During the Italian Wars, Swiss pikemen were often in French employ, and thus repeatedly ended up clashing with Landsknechts in battle. Pushes of pike between Swiss and Landsknechts became particularly infamous for being bloody in the extreme, with a combination of exceptional discipline and sheer hatred negating the inhibition against marching straight into a spiky pointy formation of doom, making them instead go full ham at each other.
