Where'd the extra 20k km come from?

From my recollections, you'll want to have the initiative set so that you move second. So B moves 5k km, then A moves to within 10k km.

The other 4 seconds in the turn

If it's still initiative dependent then doesn't the "hold position at" command have the same downside of the "move to" command?

Does the "hold position" command let Group A follow in B's movement if they are fast enough?

I don't think so, but I have never used it. I always used the follow command.

Yup, always have a high ranking officer with good initiative as the "Senior CO" of a TG, helps quite a bit, combined with:

I'd also use an order to hold position at 10k km from the target rather than closing to zero range.

You can monkey around with the Task Group Initiative setting, try lowering it and see if that helps.  I remember having this issue along time ago and I think this sorted it out. 

Does any officer bonus help with initiative? Crew training?

My 2c: this sounds like a movement initiative problem.  This is how I believe it works (sorry in advance for the lecturing tone if you already know this!):

if you look below the speed settings in the task group orders (f12) screen you'll notice that every task group has a rated maximum and current initiative. Initiative determines the move order, with (IIRC) lower initiatives moving first.

so let's say task group A, moving at 10,000 km/s is trying to close to zero range on task group B, moving at 5,000 km/s. but task group A has worse initiative.

task group A determines the location of task group B and moves to zero range.
task group B, having better initiative, notes that task group A is now at zero range and moves away at its best speed, ending up 25,000km away.

the end result is that despite task group A having a huge speed advantage, it can't get any closer than 25,000 km (the range at which task group B can move in 5 seconds.)

this phenomenon means that any weapon that depends on being 10,000km away from its target or less is kind of terrible against any kinda ship :^)

The initiative of a player taskgroup is determined by ...the ranking officer, i think. I believe AI NPRs use preset initiatives depending on their race.  If your beam ships are out-initiatived, the only way to keep the range closer than the enemy ship's speed is to split up and use flanking/encirclement tactics.

I've been using beam fighters lately and this is the problem: my fighters have subtantially more speed than the enemy, so they should be able to close to point blank range.  However, they and the enemy seem to go back and forth.  My fighters (anticipating enemy movement maybe?) go in one direction and the enemy (reacting to my fighters?) move the other direction.  I can't figure out how to beat this problem.

The best I have done is break off subsets of my fighter squad, so I have 2 or 3 groups flying around out of sync.  That way, half the time, half of my fighters might land where the enemy is going and be able to get off a shot.

Is there a better way?