Why does Tomoki never even consider that his incredibly submissive, nigh-omnipotent angeloid girlfriend might be willing and able to help him out in things beyond poorly thought-out perverted schemes? For example, during the episode where he is attempting to get a DVD, but doesn't have the money for it, instead of engaging in a Zany Scheme of selling old junk to make money, just ask said angeloid, who clearly demonstrated her ability to conjure money on demand in the first episode, to use one of those cards to get the money he needs, or just the DVD itself?
Because that's the kind of guy he is. All he really wants is to enjoy naughty things... he's not interested in power or heroics or becoming a god (either benevolent or malicious). The fact that he thinks of things in a down-to-earth way (for a comedy anime anyway) and doesn't abuse Ikaros' power (for anything worse than panty raids) are part of what makes him actually worthy to take care of the Angeloids.
Furthermore, why are all of Tomoki's perverted schemes beyond the first episode so incredibly badly planned? In the first episode, he cleverly used Ikaros' abilities to satiate all his perverted desires while escaping consequence-free, thanks to little tricks like stopping time and making himself invisible. So, why not do so again to sneak into the girls' bath or locker room? It's not like Ikaros lost the ability or will to do those things for him, if he asked.
And they're badly planned because he's a complete goofball who doesn't think things through. And again, by the end of the first episode he generally tends to see Ikaros as someone to protect and help develop as a person rather than a Green Lantern Ring to use as he sees fit.
Because the Reality Reset at the end of episode one got rid of all the 'new style' cards, including the Money Machine.
Why isn't anybody (except the Protagonists) bewildered by the fact that the Angeloids have wings and can fly around like birds, dammit!?
How long do Angeloids already exist? Ikaros at least exists already since the downfall of the tower of Babel, Nymph doesn't seem so young either, and the Harpies' and Astraea's age is totally unclear. But the second generation types seem to have existed for only a short time... So why does the Angeloid production suddenly skyrocket?
In Episode 3, Ikaros's spec sheet on Sugata's laptop indicates that she is 70,000,000 years old.
Why does the Master of the Synapse not just take Tomoki and company out with one single shot of the Zeus system?
Because Minos is bloody bored, and meddling with Downers is how Synapsians amuse themselves. Kind of like how a certain diminuitive Angeloid is fascinated with soap operas. Downers are Synapse's soap opera.
What do the chinquapin trees represent? In the anime, when Nymph comes to Earth, she lets drop a seed near the 400 year old cherry tree. The sapling from that seed grows throughout the first season. We also see 4 or 5 chinquapin trees near the Rule in Synapse when the action shifts there. In the second season, the characters visit Synapse via Tomoki's dream. The chinquapin trees are still there, but dormant - without leaves. Sugata comments on this and notes that it's unusual, since chinquapin are evergreens and can't lose their leaves.