Learn English – Talking about past events using present tense

In situation 1, which version you select (they're both grammatically correct) depends on the tense in which you're telling the story. If you're telling the story in the present tense ("So I'm walking along the street..."), you would use option A to match, while if you're telling it in the past tense ("So I was walking along the street..."), you'd use option B.

In situation 2, you'll generally want either A or C.

The essence of the grammar article was probably that in English, it's possible to discuss either past events or hypothetical future events in the present tense, as if you were placing yourself in the time and narrating from that perspective. In this case, you'd use the present-tense "first time I've been", but it's much more common to narrate this way when relating past events than when talking about future plans.

This is the correct form. The verbs get, walk, put aren't present-tense forms but unmarked infinitives (unmarked means they aren't 'marked' with to). That is one of the two verbforms^† which are permitted in clauses which act as complements to most verbs of perception, like see, hear, notice.

It is sometimes difficult to distinguish an unmarked infinitive from the plain present form, because there is only one verb in which they are different: be, whose plain present form is are.

But when, as in your example, the verb's subject calls for a 3d-person singular form, it is easy to tell the difference. If these were present forms, the sentence would read

... he saw the man gets out of the car, walks slowly along the footpath at the edge, and puts a hand on a rail.

^† The other one is the present participle; you could also write

... he saw the man getting out of the car, walking slowly along the footpath at the edge, and putting a hand on a rail.

But the participle is usually used to say what someone is doing at a particular moment, not a series of things done; so the infinitive is preferred here.