Is The HBD Stabilizer Frontrunning Everyone Else For HIVE to HBD Conversions?

No, frontrunning is a different phenomenon that you are conflating with the actual process (which for the most part you describe accurately, so I hope you understand the difference without further discussion).

so to be on the safe side you really want the price for HBD to be up a lot.

I disagree with this somewhat. An important factor being ignored here is "how long is the price staying above 1.05?". If I saw price of HBD sustained at just 1.07 or so for a few days, I'd be quite happy to convert, wait the 3.5 days, and likely profit. It would also be useful to take into account how many other people had already initiated conversions (to see how much HBD will likely be created to counter the price change).

What this means that the stabilizer is the only one that can profit for a HBD price in the range of 1.01 to 1.05.

You are correct that only the DHF can profit by HBD arbitrage using conversion mechanism in this range. But this has been discussed several times in the past (I think smooth was the first one to point it out and I think I've discussed it as well). But someone can profit from HBD just by holding it in savings or by normal arbitrage between exchanges where the price differs (this happens all the time).

Now the DHF is not a single entity and it serves the whole Hive ecosystem, but I think it still needs to be on a same level playing field with everyone else, meaning selling HBD for the same premium as the fee for conversions.

Sorry, I 100% disagree with this idea. For small variations in price peg, I think it's much better to have everyone in the ecosystem benefit, rather than a few "savvy" traders.

At the moment that will be 5% premium or more preferably we change the conversion fee from 5% to 1%. This will most likely make HBD to scale more easy as well. Maybe there is still room in the next HF for this?

The 5% premium was set to discourage someone attempting to profit from the conversion mechanism by manipulating the price of HBD (correction: price of Hive). I think it is very prudent to keep it at this level.

The point of adding the Hive->HBD conversion wasn't to make a profit for traders, it was just to prevent breaking of the HBD price to the high side of the peg. And it has proven very effective at doing so, I think, by introducing easy liquidity of HBD when the price starts to rise too far.

This will most likely make HBD to scale more easy as well.

It's not at all clear to me how this would make HBD scale better. The stabilizer is already offering quite a lot of HBD for sale at a small premium and the amount offered exceeds demand.

The HBD peg is also holding in a "nice range" from my point of view.