Learn English – Should I use “did “or “had” in this question

I think the primary source of your confusion is conflating three very similar uses of these words. “Break” and “down” can be combined in different ways in different grammatical functions to say different things.

“Breakdown” as one word is always a noun, specifically a catastrophic event that a car can have.

breakdown    noun

a failure of a machine to function : an occurrence in which a machine (such as a car) stops working
_{Source: Merriam-Webster definition of “breakdown”}

• My car had a breakdown.

“Break down” as two words is a phrasal verb, the first word of which can be conjugated to indicate the timeframe of when a breakdown occurred, e.g.:

• My car broke down near the circle around eight o’clock this morning.
• My car has broken down.

“Broken down” can also be an adjectival phrase that describes a car that has broken down, e.g.:

• My car was broken down until ten o’clock this morning.
• My car is broken down.

I called my friend but he didn’t pick up my phone. I didn’t try to call him immediately because I thought he would be/would have been driving his car.

Two things before I get to your question: First, your friend didn't pick up his phone, not yours. Second, if you're talking about calling him again, you should say "call him again immediately".

Both of your scenarios can use the same words. Neither "would be" nor "would have been" is correct. If you're only guessing that your friend might have been driving, you should say "I thought he might have been driving" or "I thought he could have been driving". The first one is more common.

Also, you don't need to say what he's driving unless there's something special about it.

Here are some improved versions of your sentences:

I called my friend, but he didn't pick up his phone. I didn't try to call him again immediately because I thought he might have been driving. (Scenario 1)

I tried to call you earlier, but you didn't pick up your phone. I didn't try to call again immediately because I thought you might have been driving. (Scenario 2)

And here are some other variants:

I tried calling you earlier, but you didn't pick up your phone. I thought you were probably driving. (More casual; implies more certainty about him driving; uses context to imply that you waited before calling again)

I tried to call earlier, but you didn't answer. I thought you might have been driving, so I waited to call again. (Shorter; uses context to imply who was called)

I tried calling earlier, but you didn't answer. I thought you might have been out driving your new car. (Focuses on what he was driving)