Let me try to answer your questions in order:

1. Was there a 'Spanish' style of armour? Yes. There are some commonalities with Italian export armour and the armour of other regions like Flanders, but yes, the Spanish seem to have had their own stylistic preferences. There were Spanish centers of armour manufacture, including Burgos, Seville, Calatayud and Castejon de las Armas. Unfortunately, very little natively produced Spanish armour survives - even in the Real Armeria many of the pieces are of Flemish or German manufacture, due to the strong Hapsburg connection in the 16th century to those two prolifically armour-producing regions.

Some interesting features visible on Spanish effigies are a kind of finely articulated mitten gauntlet that does not entirely cover the fingers - the articulation is much small than similar Italian guantlets. Like the French, English and Germans Spanish effigies display more fluting on the limbs than you generally see in Italian examples. Sometimes there is a high plackart* decorated with scalloping. This effigy of Juan de Vargas of Barcelona (died 1515) shows some of these features.

As to the sort of helmets worn, effigies depict their figures bare-headed but sometimes you see a sallet or the bevor for one among the various carvings on a tomb. Some Italian exports to Spain used armets, and by the 16th century there is a distinctive 'Spanish' armet that survives in small numbers, like this. Note that the top of the visor is nearly vertical and the front/bottom of the visor is curved and fluted in a bellows pattern. The spanish were fond of kettle-hats and sometimes wore them with bevors that extended so far as to cover the face (such that they needed eyeslits) but I am not sure if these would have been worn by heavy cavalry.

2. Did the gentlemen at arms of Castille and Aragon use imported Italian armours? Yes. The most famous example of this is the armour of King Ferdinand of Aragon, now in the Kunsthistoriches museum in Vienna. It may have been made in Italy, or by Italian masters working in Iberia. The Missaglias for one had workshops in Iberia, and Ferdinand also ruled Sicily and was quite involved in Italy in general. It does not have any of the fluting or other features noted above that may be marks of 'native' Spanish armour, or at least the native stylistic preference (which could be met by imports that more closely matched local tastes).

3. I don't know much about the military of El Andalus, so I can't comment on the differences between Aragonese/Castilian knights and heavy cavalry of Granada.

EDIT: I should not that Claude Blair identifies the kind of 'fish tail' plackard on the Ferdinand harness as characteristically Spanish, then admits it is common on other Italian armours. I am not sure, since the two examples I have seen on Ferdinand's armour and Phillip the Handsome's armour, neither of which were likely made in Spain, or at least made by Spaniards.

*(either the lower half of the breastplate, or a reinforcing piece, depending on how the breastplate is made)

I would like to add to the question: being castillian myself I know how hot can it get here in the summer, how did they adapt to that, did they fought on cooler months or was the armor different from colder climates?

Pinging /u/Itsalrightwithme

They wore both foreign and locally produced armor. There were many different variety of styles at the time but I do not believe that the Spanish produced suits that differed enough from others in the 15th century while at the same time resembled each other enough to deserve a separate distinction.

Here is what they wore around that time(a short google-fu);

http://static01.nyt.com/images/2015/05/15/arts/15iht-rartbargello-4/15iht-rartbargello-4-articleLarge.jpg http://www.gutenberg.org/files/47878/47878-h/images/ill_012_sml.jpg http://www.gutenberg.org/files/47878/47878-h/images/ill_020A_sml.jpg https://s-media-cache-ak0.pinimg.com/736x/50/08/49/5008490cc89203896fb68f0656ab9d29.jpg

"I would like to add to the question: being castillian myself I know how hot can it get here in the summer, how did they adapt to that, did they fought on cooler months or was the armor different from colder climates?"

Surprisingly, we have very few, barely any sources that tell us of heat exhaustion because of wearing armor, neither do we have any evidence that the weight of it significantly impeded their stamina or tired them to any serious extent.

They apparently just dealt with it.

Remember, people wore suits of mail, lamellar and coat of plate armor in Africa as well and they did not simply collapse either.