1)At the end of battle, the Defender have no units left and Attacker have one routed unit.

Who won the battle?

Tally strength is defender 0 vs attacker 0. Attacker retreat? Mean the region become an empty region?

2) Elf archer as attacker draw an orb, Flesh ripper as defender also draw an orb.

Elf archer take out the flesh ripper before he gain 3 health?

Since attacker orb resolve 1st.

Defender wins on a tie, a routed army is not considered, as they are running away.

Yes the attacker resolves his orb first and kills the Fleshripper.

If you want rulebook references:

Referring to page 21 of the rulebook, item number 5b states that "Each player totals the number of standing units he has in the battle." Since both units have no standing units, they both have zero strength.

Item number 6 then states that "in the case of a tie, the defender wins."

According to page 22, in the section entitled "Rounds of Combat," item number 3 states that "Starting with the attacker, each player resolves his units’
special abilities...." This means that the archers damage the fleshrippers before their ability activates.