Pudding means a lot of different things in different places. In general, it refers to savory dishes or custards (that are not always desserts - savory puddings are often part of main meals). British english refers to pretty much any dessert as pudding, and American english refers almost...
It comes from the genre. Generally in male MC otome games, your goal is to woo the girls and in female MC ones, the goal is to make the guys woo you. In the former your role is mainly to find paths to aggressively pursue, but in the latter your job is generally to find paths that passively make...
Angel Next Door is based on a completed set of light novels, so it has direction after they get together, and has a set ending. The second season of the anime will finish it out, provided they don't do After the Rain as well.
Four chapters in and this is heading towards being one of the best "villainess" titles of all time. The writing is full of heart, the story is on point, and the tropes have enough of a twist to them to keep it fresh. Well done.