Firstly I hope you're not putting jump drives on all your warships, at early tech levels they take far too much valuable tonnage, better to stick to groups of 3 ships, I usually put a pair of offensive ships with a single jump capable ship of the same tonnage but with large sensors and some point defence weapons.
Secondly, early engine tech is horribly slow, to make effective warships you basically need to use fuel guzzling high multiplier engines, and devote a high tonnage to engines, I would go with 20-25% engine at nuclear pulse. Your end up with a fleet that can only venture out of sol once every 5 years due to fuel shortages, but at least it will be effective in combat when you do field it.
I'll answer your questions to the best of my ability;
1) I'm pretty sure the precursors and swarm generate when systems are created, so your rollback should avoid them if you don't to any exploring, note however that NPRs will be exploring and possibly generating spoilers for you to play with.
2) you definitely aren't too far behind, I like the laser tech you have there, it's quite sufficient if you had more speed to outrun your enemy, but the problem is at early engine tech that's unlikely.
But fret not, as I said earlier, higher engine tonnage and more multiplier will help you catch up, and for better sensors at low tech just make them bigger. Your best bet at having sufficient offensive capability at early tech levels or any tech levels really is missiles, even if you're woefully lower tech level than your enemy you can still crush him by sheer weight of ordnance even if they're only 20% faster than his ships and suffer a 3% hit chance, you just need to be able to fire 5000 missiles :p
3) sm mode will not reveal much about the NPRs, it might occasionally give you a hint why a short interrupt happened but no actual spoilers.
Returning to your ships , 833 km/s is too slow for a warship at nuclear pulse , I think you can get 2000km/s easily, try taking that design above and copying it, design a size 25 engine with your highest available multiplier( use sm mode and create one instantly, add prototype to the name so you don't cheat and use it without researching it) then put 2 of those on each ship, on ship 1 remove the jump drive, and the hangers then add as many extra 15 cm lasers as you can, on ship 2 remove the 15 cm lasers and fire control and maybethen put much larger sensors, try a size 2 resolution 1 and a size 12 resolution 10.
You should find you end up with a pretty effective squadron of more specialised ships, though I might be wrong about if there's room for more sensors on ship 2 after adding the new engines, maybe reduce hanger space or make a dedicated carrier . Though I do like the idea of cruisers carrying a small wing of recon fighters..
