Do FAs, spire runs, etc. get harder once you unlock & start a new chapter?
Yes and no. "Harder" is a tricky word.
For FA, the coin and bracelet badges are tied to chapter. The threshold to make one goes up as you chapter up. That does make it "harder", so yes here.
For tournament/Spire, "difficulty" in general is determined by how far/deep you go. The deeper/higher you go, the wider the margin between the number of your troops you take to the field and the number of enemies you face. This ratio is the same for everyone for the same point in tournament and Spire regardless of what you build in your city. Therefore, this part never gets more "difficult" at the same encounter in the tournament/spire and it might even get
easier as you get more toys, like wonders that buff your troops, unit promotions, etc. You determine the difficulty by how much you want to do. The more you do, the greater the enemy outnumbers you.
The second factor that affects "difficulty" is the troop combination. This is somewhat randomized but with a propensity of certain troops types to appear more predominantly at certain tournaments. For example, the marble tournament is always really easy because the dominant troop type is Heavy Melee and they're the easiest troop to handle. But when you get to Dust, it's pretty much an ugly combo of melees and range enemies and it will just hurt. Dead bodies will pile up. None of this is determined by your chapter though, and therefore, in this case, it's a no.
The third factor of "difficulty" is one's ability to replenish troops before the next week's Spire/tournament. This is the one that will be affected by what you build and how many mandatory techs you unlock. The "further along" in the game, which includes things like how big your city is plot wise and which chapter you're in (mandatory techs unlocked), the bigger the stack (yes, I said stacks
@samidodamage!) of troops you bring to a battle. And in turn, bigger stacks lead to more dead troops that need replacing. This is also determined mostly by user player preference. If a player does minimal tournament and Spire, you don't have that big of a burden to create large armies so it can become negligible. If you are doing a lot of each or both, then you need to be able to make back all those dead troops in time by next week. If you are gung ho on tourney/Spire, then yes, it is recommended that you upgrade your military buildings and wonders to offset the increases so you can make back dead troops in time. I've witnessed a lot of players sprint through chapters and then can't handle the Spire anymore because their city lacked the foundation for bigger troop/catering requirements.