line 2 below ... it searches for a leter in lowercase.

Actually, it's any letter, upper OR lower, because of the /i modifier. Which makes this line 2 exactly equivalent to your 'this might work' line (let's call it line 4), so yes, line 4 works as well.

Is the letter $0 and the number $1 ? I assume so.

$0 returns the name of the program being run. $& returns the entire matched string. $1 and $2 return the first and second captured (patterns in parens) matches, so yes in lines 2, 3 and 4 the letter is $1 and the number is $2. Note the exception here is line 1, where the number is $1 -- this is because the look-behind (?<=foo) is zero-width and isn't considered part of the match.

I am curious how lines 2 and 3 below work.

Your description of line 2 (with the above case-correction) was accurate. The difference between line 2 and line 3 is that line 2 matches a single character from a character class (in this case only a letter) and that line 3 will match any character at all. For example, try them both against the strings "A1", "+1", "11", "A111" and compare the results.