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