If you cast a card "without paying its mana cost," you can't choose to cast it for any alternative costs. You can, however, pay additional costs. If the card has any mandatory additional costs, you must pay those to cast the card.

You can tap any untapped artifact you control to pay the cost of the mana ability, including an artifact creature you haven't controlled continuously since the beginning of your most recent turn. Tapping an Equipment this way won't affect its abilities or the equipped creature.

Urza's last ability doesn't change when you can play the exiled card. For example, if you exile a sorcery card, you can cast it only during your main phase when the stack is empty. If you exile a land card, you can play it only during your main phase and only if you have an available land play remaining.

The token created by Urza's first ability will count itself, so it'll be at least 1/1.

If you don't play the card exiled with Urza's last ability, it remains in exile.

If a spell has in its mana cost, you must choose 0 as the value of X when casting it without paying its mana cost.