I think it depends on the research. e.g. in Medical research you first take topic and then do the research. Generally in social science you based on your interest you start exploring literature to finalize your research topic.
What comes first? Does your university have parameters you must follow - a dissertation template? I can only answer as it relates to social science. In this case, you should have a final topic before starting your literature review. Although, the research questions can evolve a little bit as you read (in communication with your supervisor), the purpose of a narrow research question is to set parameters and to anchor your study firmly so you don't drift off and waste too much time on interesting tangents as you read.
Next in the sequence, choosing a methodology before "research" - assuming you mean data collection? Before data collection, you must choose a methodology aligned with your research objectives and one that will work within your financial, time and other constraints. Choosing a methodology requires a discussion of the pros and cons of your choice, justifying your approach to your topic in light of your objectives.
In my experience, the first three chapters (introduction, lit review, methodology) must be approved before actually conducting any 'on the ground' data collection. As well as approval from your committee, you will need a formal review from the Institutional Review Board (IRB) of a university before starting data collection. Many student projects don't require IRB oversight, but some do - and the IRB must sign off either way.
Last - you must analyze the findings objectively without commentary (chapter 4) before you write your synopsis, conclusions and recommendations (chapter 5). Therefore, although some universities do this differently and you must check with yours, I feel a good sequence is as follows.
It usually starts with course work and simultaneously you will be given some time to submit your Review of Literature (ROL) and after that your guide approves it then you have to submit your synopsis (which indes your whole structure as in ROL, Objectives, Hypothesis, Scope, Reserach Methodology, Sample Size, Theoritical Model, Expected Outcomes) and finally you will face DRC departmental research committee where in you will discuss about your topic and only your respective departmental committee will sit in the panel and in case you will be asked for any changes to make you have to incorporate them accordingly and at final stage URC will be conducted which is at higher level where higher authorities, external faculty sit and you have to defend your topic there. There in it will get either rejected or approved.
It's always good to review the research literature first, even if you have a good idea about your topic. New research should be on the cutting edge of discovery. You need to know where that cutting edge is.