To solve (1), just any old scrap of paper, which is then (2) transferred to a personal wiki that lives in dropbox. I use this one (tiddlywiki.com), but anything with tagging and search would work. I appreciate the organization by time, tag, or list, and also the latex and markup support, but these are not essential. Semantic search would be even better. Ive used this to capture ideas, notes, code, writings and lists. It has been essential for my work/dissertation.
I see the personal wiki as an evolution of the commonplace book, that was kept in the past by the likes of Darwin (https://en.wikipedia.org/wiki/Commonplace_book). Here are some of my notes from the book Where Good Ideas Come From
"We know much of Darwin's thinking on the development of his ideas from his extensive notebooks, which he read, and re-read and recombined. This era was the time of the 'commonplace' notebook, in which long passages of quotes from other sources and thoughts were recorded. Reading and writing were apparently quite related. These books may have struck a balance between silo'd organization, and utter chaos, allowing the development of theories beginning with hunches which could then be further developed. The key to developing a hunch into a theory appears to be writing it down "