you are required to put down EVERY institution whether you think it's the best story to present or not. all four schools need to be listed. now you just get to decide how to tell people about it.
i will have 6 total transcripts (4yr univ classes while in HS, undergrad univ, study abroad while in ugrad univ, grad school univ, and 2 local schools-1 CC 1 4yr - where i filled in a few prereqs during grad school bc they were cheaper than taking them at the grad school). so no it shouldnt affect you if your story as to why makes sense.
as to the research yes getting exposure to different areas is great, but no i would say what will look better for you is getting some solid time in 1 lab so you can talk about the significance of the work, the depth of YOUR contribution to it, and hopefully have a meaningful publication in the end.