I do indeed have a habit of making new things too strong and then having to tune them down.
That's totally understandable when in love with an idea, no problem!

Worth noting that you were facing ~120 DP worth of ships with officers and you still won without losses, which seems reasonable to me. The reason one of them was so mobile was because it had safety overrides.
Didn't quite notice for SO, so maybe that's how this particular cookie crumbles (but then again, SO itself may have to be rebalanced because it is stupid good ATM, eh... another can of worms).
The 120 vs 120 isn't the whole picture though, all my ships had 3 s-mods, all commanded by level 6 aggressive officers with two elite skills etc. I am really not sure it would have gone the same way without officers on these ships and without s-mods, in fact, it may have been extremely difficult. Hard to tell if situation would have been similar facing a midline line-up instead but my guess is that it would not have been THAT difficult (and maybe I'm wrong but I see ADC as being an exotic midline faction, they seem closer to vanilla midline than high or low tech to me). With that said, if you rebalance Temporal Damper it may just be fine.
Eidolon: It's a fast capital, and definitely not the only one out there (Odyssey, DME's Zelenograd, II's Dominus, etc). It's fantastic at bullying things that are smaller than it, but loses against "real" capitals- an Onslaught beats an Eidolon quite handily.
You're correct. I don't use Onslaught, just did some tests and it works quite well against the Eidolon. I'd bet Conquest would as well. I never had a problem with a Zelenograd kiting my destroyers all around the battlefield for minutes on end though (I haven't compared the stats side by side but I was under the impression that just coughing on a Zeleno is enough to break it in two, it is much less the case with the Eidolon, the latter scares me, the former not really). But it may just be circumstantial (and maybe when Eidolon is kiting like crazy there are mods/officer skills at play, just like the Komodo example).
Temporal Damper: You're correct that ships with damper field are normally just tanks, or otherwise have some tradeoff for having damper. In this case the tradeoff is that they're just overpriced for what they are- without damper, the Komodo could be ~25 DP (Dominator equivalent) and the Caiman could be ~8 DP (worse Sunder). The Apex could definitely still be 60 DP without damper, but the Paragon is a thing and only costs 60, so the Apex is staying right where it is. Regardless, temporal damper is a little bit too strong right now, so I'm gonna reduce the system uptime a bit.
Sounds good! The Apex is in a good place I think, even if Temporal Damper is brought down a notch. I always use this ship in sim as a "golden standard" to test all my Battleship builds and it is definitely stronger than the Paragon builds available there. I have never had a build beating an Apex in sim but not a Paragon and definitely had a few beating the Paragon but not the Apex. But then again, Apex is more expensive (one may argue it should be 70 instead of 65 but with a slight Temporal Damper nerf it will be fine... 5 DP is also around the cost of most fast frigates which, in some cases, is not an easy choice "do I want Paragon + fast frigate or do I want Giga chad paragon with Temporal Damper but no fast frigate"... that sounds balanced to me).
Crocodile: This one's been on the watchlist for a bit of a nerf. Will probably reduce hull HP and OP a bit. Though, it's still only good in a fleet context- it struggles to kill any other similarly-priced cruiser in a one-on-one. And yeah, it's not a particularly fun ship to captain. A bit like an Eagle, there- there's not much going on with it, so it's a bit boring to fly. But not everything needs to be playerbait, and it's perfectly fine in AI hands.
I've been flying Eagles a fair bit, that's my go to ship in vanilla and trust me, 2-3 HMG + 4-6 AM blasters SO Eagle build is very fun, it is like flying a brick to bash the enemy fleet with: not subtle at all, very stupid but very efficient.

But I agree regarding the croc, alone it is laughable, in a fleet it is a massive PITA that saturates PDs and fighters/bombers. I think all factions should have such a ship ideally (independently "trash" but a godsend within a fleet) .