Your second point is incorrect and is easily rebutted.
Suppose someone opposed driving cars, and, everytime there was a fatal car accident, claimed their position was strengthened.
This argument is invalid because people who support driving cars already know and understand that fatal car accidents will occur, and accept it as one of the risks of driving cars that presumably does not outweigh the benefit. In other words, each new fatal car accident does not provide any new information or a new argument: it is a position driving supporters have heard, considered, and reached a decision on.
No sane opponent of gun control denies that guns (knives, bombs, cars, snakes, battery acid, etc) can kill or harm innocent people, either by accident or intentionally.
They have already reached the conclusion that the harm of gun control outweighs the benefits. Each new incident does not provide a new argument.
Suppose, for example, that gun control opponents had provided a reasonable explanation of why gun control is bad after the last mass shooting.
You claim that the new mass shooting weakens their argument, but it doesn't. They can use the exact same argument as before: "we've explained previously that gun control is a bad idea despite all the mass shootings through yesterday; the exact same explanation applies to the mass shooting today, and will apply to all mass shootings in the future. We are already aware that mass shootings have occurred and may continue to occur, and explained why gun control is a bad idea despite these mass shootings".
A weaker argument would be: "the infrequency of mass shootings demonstrates that current gun control laws are adequate; if the laws were inadequate, mass shootings would be a leading cause of death, and they are not".
Another form of this argument: "if gun control laws were inadequate, over 1,000 people would die daily from mass shootings; the fact that this mass shooting killed fewer than 1,000 people AND that there has not been mass shooting deaths of 1,000 people daily, proves that gun control laws are sufficient. Each mass shooting reminds us how rare mass shootings are, just like each airplane crash reminds how safe air travel is (the media doesn't report on trivial events). Therefore, this shooting is actually proof that gun control laws are adequate".
Your first point is interesting and resulted in a long debate between me and @jobermark which doesn't answer the question, but may be of interest.
To summarize, I would say appeal to emotion in a moral argument isn't a logical fallacy per se, provided that we accept morality is ultimately based on emotion, provided that you don't make any other logical fallacies while employing the argument.
In your "war is bad" example, you are making a one-sided argument (you're not providing emotional arguments that living under oppression is worse than death, for example), but not one that is inherently illogical. Here's how I would breakdown your logic:
You: War is bad because it kills people
Opp: I accept that war kills people, but don't accept that killing people is bad.
You: What if it was your family that was being killed?
Opp: Yes, it would be bad if my family were being killed.
You: Do you believe your family is morally superior to all other people?
Opp: No, there are other good people in the world. Therefore, if it is morally wrong that my family were killed, it is also morally wrong that other people (not all other people, but at least some other people) are being killed. I recant my earlier position that killing people isn't bad.
You: Therefore, you accept war is bad because it kills people:
A: War kills people (you accepted this statement originally)
B: Killing people is bad (you now accept this statement)
Therefore, C: War is bad.
Opp: This argument is invalid because war does many things, not just kill people. Additionally, although I agreed killing people is bad, I didn't agree that anything that kills people is bad: it's quite possible something kills people (bad) but also has benefits (good). You have convinced me that killing people is bad, but not that it is a bad that can not be mitigated.