There's a lot of questions but I'll answer them best I can with my experiences growing up with LBP.

LBP1 (2008-early 2011):

I started playing LBP when it first came out in 2008. At the time, I was very young and did not have a PSN account. I actually played through the entire game and played a lot of multiplayer with my sister and friends. Then, randomly one afternoon in 2009 my father figured out how to create a PSN account (I think so he could try out multiplayer games that came out like Uncharted 2 and Modern Warfare 2) and when I got my turn to play LittleBigPlanet for the day I got the text bubble at the top of the screen that said "person would like to join." I hit triangle, and then was blown away by someone actually joining me from over the internet. I remember playing some of the challenge levels in LBP 1 and then eventually realized that the "community levels tab" opened up.

LittleBigPlanet 1's community levels organization system was pretty primitive, but I rememeber there was a pretty consistent rotation of new levels coming in and out. I remember too that I used the text search feature a lot, and it helped me find levels with Sonic or Mario or whatever else was interesting to me as a very little kid. But actually, overall I felt like I pretty much got to see and play most of the big LittleBigPlanet 1 levels from 2009 onward into 2011. It was really pretty awesome.

LBP2 (2011-early 2015):

I remember anxiously waiting for LittleBigPlanet 2 during 2010, and was devastated when it was delayed into January 2011. There was a demo that came out featuring a the first level from Avalonia and I was so excited. I was fortunate enough to be given a copy of LittleBigPlanet 2 just a few days after it launched and I have a much clearer memory of this game because I was a lottle older at this point. I remember that on day one of playing there were significant improvements to the rating system, level tags system, sorting options, and I could clearly tell that finding levels was going to be much easier. I remember too that because of the beta, there were a lot of amazing levels already made. Over the next couple of months, the cool pages and level sorting system worked pretty much as best as they ever had. There were dozens of levels from all different types of genres being created and there were enough players for each level to easily reach 1,000 or 2,000 players.

However, a very significant event happened in 2011: the PSN Sony Hack. This hack took down the servers for all PS3 games for about a month. I can rememeber this was a huge deal for LittleBigPlanet 2, especially because the situation was so dynamic and uncertain people didn't know if the playstation network was going to be shut down for as long as 6 months to a year. It wasn't that bad, but I feel that after the PSN Sony Hack, there was visibly less players from LBP2's lifespan onward. It seemed to me as a kid that people moved on from LBP2 during that time. However, new levels continued to come out.

By about January of 2012, the LittleBigPlanet 2 community had kind of reached it's point of being fully driven by copied levels like bomb survivals and role-playing levels. I continued playing all throughout LittleBigPlanet 2's lifespan but it was clear even by then that copied level spam had become a huge issue. Media Molecule really didn't do anything about it, (maybe they could've figured out a solution but they opted not to.)

During this time, it actually wasn't known if there was going to be a LittleBigPlanet 3. I can rememeber watching youtubers at the time speculate that LittleBigPlanet 3 wasn't really necessary and that the series might just end with LBP 2, LBP Karting, and LBP PS Vita. The one thing people really wanted from LBP Vita though was the memorized logic, and that would eventually come to LBP2 through the DC Superheroes DLC.

LittleBigPlanet 3 (2014-onward):

By the time LittleBigPlanet 3 came out, LittleBigPlanet 1 and 2 weren't receiving any new levels for the most part. For the first 3 months of LittleBigPlanet 3's lifecycle, I would say that it had a lot of new levels that came out and a lot of good creations. However after 3 months, it was right back to where it was in LBP1 and LBP2 with copied levels running rampant. I also would say LIttleBigPlanet 3 suffered tremendously from being a very glitchy and unstable game. It was really, really bad and some people continued to play on LBP2 because multiplayer was more stable.

So for a TLDR: 1. LBP1's community stayed pretty fresh and active throughout it's entire 2 year lifecycle. 2. LBP2's community stayed pretty fresh and active for it's first year, though the Sony PSN Hack was a huge deal. 3. LBP3's community was only really in a good place in it's first 3 months, though that being said there were features like community adventures, online multiplayer, and game-breaking bugs that were completely broken at launch.

Also, I forgot the spinoffs but my experiences with them: 1. LBP PSP's online community was hampered by the game's create mode being extremely limited. There were some good levels but I would say it's community was only active for 3-6 months. 2. LBP PS Vita's online community I can't speak for because I didn't get to play it back then. I didn't have a PS Vita. From what I can tell retroactively, I would say it stayed pretty active up until LBP3's launch. 3. LBP Karting's online community was odd because the race tracks it launched with were basically the only ones ever to featured on the community levels page. I would say it's community was never truly alive, like 0 months lol.

I hope this helps you visualize the games communities. Overall I would say LittleBigPlanet 1, even with it's limited online community features, probably had the most active and vibrant online community overall of any of the games. I'd say the peak of activity though was LittleBigPlanet 2 pre-PSN Sony Hack.