I believe that you are looking at a mismatch in the meanings of tenses between English and Spanish. Not unreasonable, DL decided to teach the present tense first. You use it all the time in Spanish to talk about what is happening now and some future events. However, in English the present tense is NOT used to describe what is happening now, but instead is primarily used for habitual actions and for describing facts that are always true. To talk about what is happening now, English uses the present progressive - a compound tense. Unfortunately, the tenses don't seem to line up and DL appears to want to match present tense to present tense in its translations. So, just as you said, "She is forming an institute.", is a better translation.
Having used DL for a bit longer now, I see that DL covers first the present tense with the present progressive (the -ing form) coming in later lessons. Until further notice, and disregarding what "sounds right" in English, translations should avoid the -ing form - this comes later here in DL land. This is the kind of thing DL could easily make clearer with some written discussion of what is being taught, but they have decided against that sort of thing.
I agree that "Form" is a strange enough word choice in English for the act of establishing a school/institute/etc that this should be "She founds an institute."
I'd say that "they form a school" means that the school is collectively composed of "them" in English, not that "they" were responsible for creating the school initially. For instance I would assume "The United States, Mexico, and Canada form NAFTA" to mean those are its current members, rather than that they started it (though in this case both are factual).
From this forum posting: http://forum.wordreference.com/showthread.php?t=80638
An institution is a large building, typically where the mentally ill or orphans live and are taken care of.
An institute is an organization that exists so its members can do a particular kind of educational or social work.