In the short "Prediction", Peter said that he can't do anything a spider can. Then the seer asks if he can get her out of the web. Thing is, spiders can't get things out of their web! Technically they can, but most of the things that get trapped are its food.
The burglar's gun being blue doesn't just have to be a way to make it 'kid friendly'. In the real world, guns can be an odd color because of 3D printing. So perhaps while the bullets were regular bullets, his gun was a fully functional 3D printout of a gun.
Spider-Man is shown to not have his Hero with Bad Publicity status like most other versions of the character, with people being merely annoyed with him at worst. Why is that? Well, considering that social media is on the rise at this point in time (2017), his heroics become much more well known to the public. All of his heroics will be captured on video in some way.
On top of that, even if J. Jonah Jameson were to do his usual smearing of Spider-Man, it could quickly be disproven with social media, and even anyone who believes him would only do so for a little while.
Season 2, however, slightly dashes this statement so far.
Actually, it seems to be implied that even the mainstream press is more-pro superhero, as Peter has directly saved major media figures like Robbie Robertson, who seems to respect Spidey right off the bat.
Related to the above, while Peter Parker isn't exactly popular, he is more well-liked by other people. This is most likely because of a few reasons:
1. Because of the show being more pro-science and pro-education, people are much more accepting of his intelligence, only being annoyed when he messes up greatly in some way, is late with something, or makes corny remarks at the wrong time, and even then, no one is that annoyed.
2. He has not encountered someone directly antagonistic in his life. Anya is tough on Peter in trying to get him to be more serious at school, but she can also see when he's doing a good job. Otto Octavius, at the moment, is simply blunt and upfront, and not screwed up in the head. Flash Thompson is revealed to have pulled a Heel–Face Turn prior to his appearance in this series, and while he's still slightly mean to Peter, it's relatively harmless stuff like noogies, and they do at least try to respect each other now.
3. This is also due to changing standards, and how both schools are different. Because of zero-tolerance laws in bullying nowadays, pre-Heel–Face Turn Flash could not exactly get away with half the stuff he did. Plus, consider the standards of Horizon High—it's a magnet school for gifted and intelligent teenagers, meaning that there would be higher standards to live up to, and thus greater expectations to maintain good behavior and composure. While Midtown High is a good place to attend school, it's not uncommon to see bullies like Kevin Wyatt despite said bullying being toned down, because it's a regular school with lower standards.
Many people take issue with Peter attending a school for gifted students, feeling it undermines his underdog status. It actually makes sense when you think about it; someone as obviously intelligent as him would fit right in with other gifted students. At the very least, Peter should be given offers to go to special schools. It's also made clear that even if he's surrounded by people who are of roughly the same intellect as him, he still has his own challenges to face, like the higher tuition and increased expectations.
With the release of "A Day in the Life", this series' interpretation of Venom makes a bit more sense in why it wants to bond with Peter. He SAVED IT from being robbed. It's merely trying to repay the favor.
Why was Anya not at the Halloween party in "Halloween Moon"? Well, "Party Animals" shows that she's not that big on social events, or at least school dances, and considering what happens later in the episode, who could blame her? Miles's more impulsive personality is also a key factor since he could attract too much attention.
Conversely, it makes sense why Harry and Gwen would be at the party. Given what happened in "Party Animals", where he was forced to stay at Osborn Academy while she witnessed her uncle, who's revealed to be the Jackal, be arrested, and Peter comforted them in the end over it, they would need some way to alleviate things. On top of that, Gwen has become nicer and more sociable at this point in the series, while Harry is the son of Norman Osborn, who's big on attending social events.
After Harry finds out that Peter is Spider-Man in part 4 of "Spider Island", and he breaks off his ties of friendship upon Osborn Academy being destroyed in part 5, he does not reveal Peter's secret identity despite having ample reason to do so. Why? It's because Harry just ultimately wants to do good even with his personal issues, in order to make his dad proud. This can be observed in other episodes: in "Halloween Moon", he works with Spider-Man and Gwen to help take down Man Wolf and turn everyone back to normal. In the following episode, he reluctantly gives Peter his gauntlet to help Spider-Man take down Randal Macklin. Even in "Spider Island" itself, he's at least trying to work with Spider-Man after finding out his identity, not to mention the fact that he saved his dad beforehand. Revealing Spider-Man's identity on a large scale before his acceptance would ultimately do nothing since he would be labeled a menace by New York (or at least something bad) and this would actually hurt Norman's reputation since he's kept his hatred of Spider-Man a secret from the public, and is still respected because of it. This is best shown in the Norman Osborn fight in the season 1 finale, where Harry never says Peter's name in front of his father, since he knows that Spider-Man would be seen as a menace if Norman found out.
In "Goblin War: part 4", Adrien Toomes uses the Goblin Mech to take the control of every machine within his vicinity... except the Spiders' masks and web shooters. Why? In "Spider Island", almost everyone in New York began developing spider powers, with organic webs. Adrien just assumed the Spiders also have organic webs (which is half-true). He might also have thought the masks were just to conceal their identities and weren't filled with tech like comms or digital analysis.
We all knew this was Foregone Conclusion and Peter wouldn't go out with killing the burglar, but considering his very angry tone and how he wanted to hear that jerk's screams, Peter would've had every right to kill that bastard in.
Spidey's world is pretty disturbing. Kids are creating robots and weapons in a school. No one gives a hoot that a crazy TV show hunter is committing kidnapping, assault, and attempted murder on live television.
