The short answer is, yes, an ongoing lawsuit has a significant chance of impacting your admissions.
Assuming that your letters of recommendation know about your on-going lawsuit, it will be in their best interest to at least mention this in the letters because it is those professors who must put their reputations on the line when they draft you that letter. But let's assume that your letter writers are oblivious and they do not know about your lawsuit and you keep it hidden from the admissions committee, then what happens?
If you omit the lawsuit against your current institution, and then your new institution finds out about it, you will have (in my perspective) lied by omission and the faculty's decision to admit you has been tainted by a lack of very critical information about you which could lead to your termination from the program. Imagine you have two students, exactly the same in every single way, except that one is pursuing legal charges against a university and the other is not, who do you think the admissions committee would give a spot to?
Consider the financial ramifications as well, can you successfully pursue a graduate degree if you lose the lawsuit and now have to pay for the lawyers of the university you are hoping to sue? Can you handle the time commitment that litigation might take, and given your previous posting history, can you successfully handle the stress that litigation would bring while you are trying to do your best in a stressful academic program? Keep in mind, you're going up against a university that can ultimately flip this back on you and sue you if they wanted to.
Your best path forward is to be upfront about the lawsuit in your Statement of Purpose/ Letter of Intent/ Letter of Interest. Hiding your on-going lawsuit is, in no way, an optimal strategy in the long term. To restate the answer to your question, because I think it bares repeating, yes, engaging in litigation against another academic institution can hurt your chances of admission into another institution.