#1: After the first transformation, you can just feel the joy when Nobuo and Yumeria are shouting about how awesome it is that they've transformed.
#2: Although it does come at the cost of Nobuo being depressed, Ryuuji Sainei's cameo is pretty heartwarming that even if Nobuo is a 29-years old otaku obsessing on the aimed-at-preteen-kids Sentai, he still thanks and bows at him for being a Dekaranger fan rather than running in the opposite direction.
#5: Yumeria's mother is Dead All Along in #5 as she died five years ago, but the Grand Delusion allowed Yumeria to see her once again. Even more heart warming when you consider that the Grand Delusion syncs delusions together, so in a way, it's really Nobuo and Mitsuki who allowed Yumeria to see her mom.
#10: After hearing Hiroyo and Doctor Z's backstory, it is pretty heartwarming to see all three Akibarangers deciding to support Hiroyo all the way and condemn Z.
Same episode and could be an in-universe version: After being beaten by Delu-Knight, Nobuo got a call... from Sayaka. Who thanked him for saving her despite his Otaku origins. Admit it, not only your heart is heartwarmed, Nobuo's heart is as well, after that daycrap beating...
#12: Even if it's triggering the final episode flag and it was the producer's scheme, it is pretty heartwarming that the Akibarangers will still let Doctor Z redeem himself and reconcile with Hiroyo.
#13: Yes, they have it at the recap final episode. Though, it's at the end credits song, where the lyrics changed into the joys and freedom of being unofficial. When you think about it, Nobuo and co. are lucky they're in an Unofficial Sentai, because it grants them freedom. And yeah, that one's Saburo Hatte's fault too (a good fault, nonetheless).
The second season's recap episode. In the end, we find out that Nobuo, Luna and Yumeria really died. However, before departing, Nobuo gave out one reassuring statement: if Hatte Saburo wanted the show to have a third season, they'll still be back, via Hatte Saburo thinking up excuses to bring them back. Well, he hasn't, but probably it could be for the sake of not degrading into Sequelitis, the Akibarangers have firmly seated themselves into the heart of the fans regardless.
The three being fetched by other Sentai who have died. In a way they are recognized by the Sentai they idolize
The Akibarangers' mini shrine being made by General Pain.