Triggered abilities from cycling a card and the cycling ability itself aren't spells. Effects that interact with spells (such as that of Cancel) won't affect them.

Spellpyre Phoenix's last ability checks the entire turn. Cards you cycled before Spellpyre Phoenix was put into your graveyard count.

Some cards with cycling have an ability that triggers when you cycle them, and some cards have an ability that triggers whenever you cycle any card. These triggered abilities resolve before you draw from the cycling ability.

If you somehow cycle a card, return it to your hand, and cycle it a second time in one turn, Spellpyre Phoenix's last ability counts each time you cycled it and returns to your hand.

You can cycle a card even if it has a triggered ability from cycling that won't have a legal target. This is because the cycling ability and the triggered ability are separate. This also means that if either ability is countered (with Disallow, for example), the other ability will still resolve.