No, it doesn't have to agree with recipes, and it doesn't have to be plural.
The form פסטה is used also as uncountable or group noun. פסטות can mean multiple types of pasta, or even multiple dishes of the same type (in a non-pasta restaurant that has one dish of pasta, and it's known to the employees as just פסטה, they'd probably say שלוש פסטות לשולחן מספר חמש!).
However, in both these usages you can also use פסטה. In fact, I'd consider פסטה a bit more natural. I'd tend to say אני קורא מתכונים של פסטה (meaning different types of pasta).
In a restaurant or a cafe it will almost always be round, rather large, but thin, so about a full meal for one hungry person.
In dedicated pizza parlors it varies greatly. Some make round ones to order, some cut triangular slices out of round ones, some cut rectangular slabs out of very big trays, and you can decide on the size; You always see in your eyes how it works before you order. I never heard חתיכה used when ordering (it will definitely be used if you ask your friend at the table for a piece of her pizza). There isn't really any word for a full round pizza (a pie) - you'll just say פיצה. If you really you need to clarify, maybe פיצה שלמה. The word פאי is used in Hebrew, but not for pizza. For a slice, we used to say משולש, but I think in recent years you more often hear, alas, סלייס.