Yes, it should. Every other unit in the game can be countered on open ground by an unpromoted unit. Give it defensive bonuses and it wins against all units before crossbows. The extra hammers required to build the praetorian are not enough to make multiple axemen a viable counter without encurring a greater economic disadvantage on the defender.
As for the concept of properly promoted axemen, non aggresive civs (that is, the majority of civs) don't get access to newly-built shock-promoted axemen until they've gotten a great general, feudalism, or theocracy. Facing a war against the Romans there's not going to be a lot of axemen surviving to get to shock. Combat I axemen only have a 50% odds of winning until against unpromoted praetorians in the open ground.
Option 3 is not viable. The immortal works with a 50% bonus against archers because the basic chariot is not designed to be a city raider. The 50% bonus gives it added flexibility - allows it to be used as a city raider. Giving any swordsman variant a bonus against archers is not a good idea.
Option 4 is more overpowered than any unit in the game. Numidian mercenary aside, there is no unit which could take out such a unit with ease.