Can I initiate my first challenge with a 1 strength character (that will become 0 strength)?
Yes. A character's strength is not a factor in whether or not a challenge was initiated. You can even initiate the challenge with a 0 strength character, then "pump" the character with an effect such as Margaery Tyrell during an action window before the strength is compared is to determine the winner at step 4.2.2. in the flowchart on page 25 of the RRG.
However, you only collect the bonus for unopposed if you win the challenge, and you cannot win a challenge, attacking or defending, with a total challenge STR of 0 -- no matter what your opponent counts.