This is all set up in no particular order.
By the rules you've given, birth rate is 0 for the duration. I can see the parents of babies and young children going mad first (100 years and still changing the same baby's nappies because it never ages), followed by anyone whose project or skill takes longer than the loop. You'd need a system where everyone took turns to be caregivers to the aged who don't die and the very young who never grow up. Unless you want to just kill them all, corpse clean up will be an increasing job every cycle anyway.
Nobody is going to want to do the same thing day after day for 100 years.
Knowledge can develop, but practised skills are possibly reset on each loop.
Everything built or consumed is reset on each loop.
Technology and industry have no requirement to be clean, resources become unlimited because you have the ultimate recycling system.
Criminal justice is going to be interesting. People should be released on subjective not objective time, but the records will all be reset each round and everyone will reset to prison/freedom as appropriate. People will have to be responsible for remembering who should be in or out. Is there crime other than murder or warmongering? Everything else is a temporary inconvenience.
I can see a tendency towards extreme socialism resetting all possessions to common ownership at the start of each cycle since nothing carries over. The other option is a drift towards "Mentats" where specially trained people act as the memory and banking system of a nation, allowing certain systems, including finance and property ownership, to continue "as normal".
I'd say 5 years.
It's the electoral cycle in this country so you could start each new cycle with an election. At the end of every cycle all the damage your government has done is reset and you get to pick a new one for the next round.
It allows the 10,000 hours required to perfect a skill. This means that you can start a new job at the start of each cycle, have time to get really good at it or really bored with it, then reset for the new cycle.
It's also enough time to develop friendships, have relationships etc. which will assist with the general mental wellbeing of the population.
It allows enough time to do a masters degree from a standing start. Education is one of the key things that will carry over, this will be key to the continued development of your civilisation.
You'll still get the steady mental breakdown of the increasing number of people who, every 5 years, wake up next to their dead partner and have to go through it all again. The longer the cycle the better.
I've only just thought of food. Given that the ageing rule would apply to plants and animals too, you're going to have to be really careful about animal/fish/plant stocks or you'll be on a diet of Soylent Green pretty quickly.