No offense intended, but getting a gun and trying to shoot it out with the thugs is a very very bad idea. It's the lorelei family that are extorting him. They are very wealthy, highly organized, well known for their brutality, and pretty much own the city. If they wanted Euca or Hershal gone, nobody would be able to stop them. You wouldn't even be able to go the the police to help you, because they probably own the police too. Jenny realized they were in over their heads, and had the right idea about going to Morse for help.
As for the remnant, finding the guy who wished it into existence and having it unwished is also a problem because you are right. It was Euca who wished it. The remnant is really just a corrupted form of Echo. He keeps resisting his core, or his purpose, and the remnant is the end result of him doing so. If Echo won't tell him what the wish was, he can't undo it.
Halley couldn't kill it no matter how much she wanted to, but it could definitely kill her quite easily. The way I understand it is that Halley draws her power from Jenny's health. The more things Halley does, the worse Jenny's health will get. According to Cecilia, this trade-off means Halley is immensely powerful for a wish, but she still has a limit. Echo on the other hand is somehow related to the stars. He seems quite weak most of the time, but there's no limit on how strong he can get. Halley acknowledges this when she says fighting Echo during the meteor shower would have been a suicidal move for her.
The remnant is a compounding problem. If not taken care of as soon as possible, it will continue to grow stronger as Echo grows weaker. Halley and Cecilia can drive it away sometimes, but Cecilia's only purpose is to defend Sissel and she might end up fading away eventually. The more Halley tries, the more Jenny will pay the price for it. Once Jenny is gone, the remnant can do whatever it wants and nothing will stop it.