Mathematics GRE

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I think your tutoring + research experience, + taking grad. courses (in any areas) is great. Many schools don't require the mGRE, and in my case I didn't submit it and got into top 20 programs, so it isn't necessary by any means. Also, not having done much algebra is fine as long as you've done some significant coursework/research in your area of interest. I think you should apply to PhD programs this fall and see how it goes, at the same time applying to more Master's programs if you want, and then choose whatever you're happy with. This is only my (very limited) advice, as someone who did PhD applications this cycle.