I would tend to agree with your characterization of tone as informal (only when speaking anyway), est-ce que being fairly standard/neutral/common these days, and the inversion being more formal – although as you note, there are expressions where the inversion is natural.

At least for the first two examples you give, I would see nothing wrong with the way you asked the question. Even the third example feels just fine to me. To insist on saying pourrais-tu instead is a bit much, if you are between friends 🙂 But some people like to use that inversion even in everyday life. That inversion is not always perceived as completely formal, but certainly always as "correct/more correct" than est-ce que. It depends on the people and situations.

Note that even with pourquoi, comment …, we don’t have to use the inversion in spoken language:

Pourquoi est-ce que tu ne viens pas?

Comment est-ce qu’il va?

sound very informal, maybe broken to some, but you will hear those all the time in familiar conversation.

What your friends recommended seem overly formal to me. I see three options:

• They’re old or work at the Académie Française
• Your tone is not quite right and it makes it hard to understand if you’re asking a question or not, so they recommend you use inversion for no ambiguity (but that’s not what I would recommend at all)
• Or… maybe they’re Canadian? It’s much more common for French speakers in Québec to use inversions, is that it? They often say stuff like “As-tu vu … ?“. Spoken French, especially spoken by young people, can change a lot between France and Québec.

If it’s not any of that, I don’t understand. I agree completely with your first part, it’s normal to find that weird, I would disagree with them if they corrected someone with these arguments.

Orally:

• I would use intonation with anyone, including the leader of a country
• I would often use “est-ce que”
• I would rarely use inversion

In writing (e-mails, SMS):

• I would only use intonation with friends or family
• I would often use “est-ce que”
• I would rarely use inversion