How accurately could ancient soldiers/commanders maneuver during battles

Potentially interesting source: https://en.wikipedia.org/wiki/Military_strategy

As this seems to be for a game I'll stick to some (hopefully educated) guesses.

Is it at all accurate to divide the army into units of specific sizes, which acts may be thought of as one (making any games much easier to make), or would the entire front line melt together to one large mob of people? (and where units such as the Roman cohort thus merely a logistical tool rather than a combat unit).

Neither or both insofar as I'm aware. More or less like today, you'd divide your army into units, each with subunits and sub-subunits etc., and basically have each level assign a goal to the level beneath it, with a caveat.

One difference in medieval times and earlier is that the units weren't really considered autonomous. Giving high autonomy to units came during WW2 if memory serves me well (it was a Nazi/Blitzkrieg innovation). So you can kind of think of earlier units as basically not knowing what to do unless a courier or some horn signal reaches them or the pre-agreed battle plan made what to do explicit, with some variance for stubborn/brave/silly leaders.

During a battle and lacking any significant landscape features, How accurately (if at all) would a commander in ancient time be able to tell a group of their soldiers to move a specific distance or turn a specific angle?

As you pointed out they relied on horns (and couriers), so not very.

During a battle, lacking any significant landscape features, how accurately would these soldiers have the ability to judge said angle or said distance?

Inaccurate. You'd likely aim at some landmark.

would am ancient commander during battle be able to tell a group of his soldiers to attack (or aim for) a specific part of the enemy army? (for instance, tell their archers to aim for the place where someone spotted the enemy commander)

Likely, with the caveat that like today taking out the main commander just means someone else takes over as commander and it's likely the troops wouldn't notice the difference.

would am ancient commander be able to tell a group of his soldiers to move to notable landscape features (trees, houses, hills etc., and how notable would these features have to be)

Be wary of a possible premise here: namely the assumption that the commander has some godlike ability to see the battlefield from above. They did not. If there's a feature behind the hill ahead of you, you simply wouldn't be aware about it unless you've sent scouts who told you they're there.

Assuming the commander is aware of the feature, then yes of course.

How accurately would an ancient commander be able to tell a group of his soldiers to move around relative to their own army (for instance, move units from one flank to the other to prevent encirclement)

Hopefully one of our Roman history experts will chime in with sources rather than guesswork, but that I'd surmise likely. In ancient times battles were very much about encircling your opponent and avoiding to get encircled yourself, with cavalry thrown in to flank lines, wreck them in case of contact, and slaughter them in case they retreated. Getting flanked by infantry likely was a good reason to pivot and build a shield wall; getting flanked by a roaring cavalry charge was a good reason to not stay around to be on its receiving end.

I have taken part in, and commanded LARP-battles (stop laughing! :-), with swords, shields, lances and archers, and can relate that for an observer, any meter of elevation is a real revelation. Commanding is so slow, and non-realtime, though, so all that insight is mostly for posterity. Troops need to be instructed beforehand, and given if-then types of orders relating to landmarks, enemy troops, times, and horn-signals ('crush archers, but do not advance beyond that treeline, retreat on hearing five blows, ...').

Landmarks always were a major factor. Hills, treelines, brooks - they all give a point of reference while also being tactical faultlines.

Scouts are invaluable, even their absence ('all scouts going south did not return') is information. But their info gets more outdated the farther they roam. Velocity of Information distribution is a major factor.

If you want to go for realism, do not show the battlefield, but show the battlefield as the commander imagines it. 'Nobody told me there was a cliff there!' plop cliff ---- 'what do you mean Carl and his troops are late? They should have been hacking at the flanks for the last hour! Where are they?' Re-strengthen enemy flanks, remove Carl's troops --- 'i've had no reports from our troops waiting in that grove all morning -send a runner to investigate!' grey out troops in grove