You have a lot going for you. Kudos! A couple thoughts…
Add Carnegie Mellon to your list. They are #8 in EE and they are also strong in data analytics (I only mention that because of your data analytics internship). Of course, they are tops in CompSci (again, I mention this because of your IBM computing course). CMU has a policy of having gender parity in all STEM majors, which greatly benefits female applicants. Because of this, you may be admitted by CMU when you are rejected by lower ranked programs that do not have a similar policy.
Take practice tests and get coaching to get a strong SAT, especially the math component. CMU’s midrange for engineering acceptances is 780-800. CMU is a little crazy with their high SATs (their midrange for humanities students is 760-780, and CompSci is 800-800), but strong engineering programs want to see a high math SAT.
I think Purdue and Rose-Hulman are more targets than reaches for you, assuming you have strong SATs. My son was admitted to both for MechE with 770M/750E, 3.9UW and STEM APs and electives. Like you, he did Robotics and SO. Note that they are very different schools in size and culture. RHIT only has 2,200 students and Purdue has 35,000. Rose focuses almost exclusively on teaching undergrads, and the professors self select for teaching over research.
You may also want to look into RHIT’s Catapult summer STEM program, the application deadline may have passed. RHIT does not lock you into a major until late sophomore year, so you can apply as a EE, but change your major. They offered our son merit aid equal to 50% of tuition. In contrast, Purdue makes you apply to general engineering, then apply for your desired major, which, depending your grades and availability of slots, you may not get.
For safeties you may want to add Case Western Reserve. Not as highly ranked as some others, but nice campus, great culture, and there are options to take humanities classes. Like Rose, CWRU does not require you lock into a major until sophomore year, and they encourage double majors and minors. When we toured, our guide was a MechE major with Econ minor. If you like building things, they have the nation’s largest build space in the Sears ThinkBox. They are also generous with merit aid.
I would not worry about any of the schools mentioned here being LGBT friendly. I think with a very small number of exceptions, campuses are pretty open.
You have a lot going for you. Good luck!