What happens if I teamed up with a player who AFKs?
If WE pair you with someone who AFKs, it will not hurt your Rank progress. This means solo or duo-queuing (where it's the third who AFKs). If you make a party and your party member AFKs, you are not forgiven.
How would the AFK Forgiveness work when solo is matched with duo, and one of them AFK?
If your party-mate AFKs, you are not forgiven. If the person "we" added to the party AFKs, then it would be forgiven. Essentially, if it's "our fault" then you won't be punished. If it's from your party, it's your responsibility.
If you pair me with someone who abandons the match, that match will not hurt my Rank but hurt the AFK player's Rank, right?
Correct. The person who AFKs is always punished, in multiple ways.
If someone from my party abandons the match, and we still win the match, will we gain rank?
AFK Forgiveness only prevents you from being hurt by the result. If you win, you'll get the progress, so long as you're not the person who AFK'd.
If I win against a team with an AFK member, will I get full Rank increase or will it be reduced?
If you win, you get full credit.
How will Rank Forgiveness work for the team without AFK? Will they get reduced rank changing effects or same as usual?
AFK Forgiveness is only relevant when your team suffers and AFK. Does not apply to other team.
What is the condition for the teammates to not be have their rank penalize? Surrender or Full defeat?
AFK Forgiveness will impact any defeat where AFK Forgiveness applies.
What happens if an AFKer rejoin the match?
If the game has registered the players as an AFK, then AFK Forgiveness will apply even if the player comes back. You know when this happens because the game says "ally has abandoned the match" and the Surrender clock changes.