What are existing alternatives to metamask with that works online inside existing browsers, that don't require installing a browser extension? What are some of the tradeoffs between them?
Something that may be of interest is https://squarelink.com/
Other than that, metamask is ahead of everyone else as far as I know.
I'm not sure what you need exactly. Uport will be a good alternative if you want to work in mobile browsers as well. But users need to have there mobiles with them to use the dApp.
if only to use in PC browsers I recommend Metamask. But with mobile browsers metamask is still in early stage. Check this post.
This list might help you as well.