Learn Dutch: Zullen we zelf een kilo friet bakken? - Shall we fry a kilo of fries ourselves?

In Dutch, when you use a measure word like kilo, liter, or glas, you place the noun directly after it without a preposition. You do not use a word for "of". This grammatical feature is exactly what this lesson covers. So, "a kilo of fries" simply becomes een kilo friet.

Why is friet singular here instead of a plural like "frieten"?

While you might occasionally hear a plural form, friet (just like patat) is almost always used as an uncountable mass noun in Dutch. It works exactly like "rice" or "sand" in English, so it stays singular even though you are talking about many individual fries.

Could I use Gaan we instead of Zullen we to mean "Shall we"?

Not for a suggestion. Zullen we is specifically used to make a proposal, just like "Shall we" or "Let's" in English. If you ask Gaan we zelf een kilo friet bakken?, it changes the meaning to a factual question about your upcoming plans: "Are we going to fry a kilo of fries ourselves?"

What is the exact function of zelf in this word order?

The word zelf means "ourselves" here. Placed immediately after the subject we, it emphasizes that the subjects will do the action personally, rather than going to a snack bar to buy them ready-made.