Xilinx university program terms issue
I am not a lawyer.  This is not legal advice.  It is how I would look at this agreement, and reflects how I might evaluate the situation.  I have only partial information, and context always matters.  I read many license agreements, NDAs, and other matters that affect IP.

Section 7 doesn't seem to cause a problem.  If you disclose anything to them related to their stuff, they can use it for anything, with anyone, forever.  But, you have no obligation to disclose anything.  If you don't tell them, they can't use it, and you can't sue them for misusing it.

Section 8 could be more of a problem.  The question if how is "relevant" determined, and it seems vague.  Unlike section 7 where there is a full enumeration of all the cases, section 8 is unclear.

In any case, both you and your institution's legal department should be comfortable, and you should have guidelines from your legal department showing how they would interpret the agreement, and what they expect from you as you operate subject to them.

If this is a large enough deal, your institution should support you and may negotiate alternative terms with the vendor.  If the deal is too small for the legal team to engage, then you may want talk with the vendor about changing, or even eliminating, most of the agreement.  Some parts are likely to be non-negotiable (such as limits on their liability, and probably section 7).  Some parts (like section 8) they could be willing to strike.

Section 7 protects their critical interests.  It prevents a customer from suing them for IP infringement, be it intentional, accidental, coincidental, or in any other way.  Section 8 helps them improve their offering.

Whatever the final agreement, if there is any agreement at all it will probably require review from your institution.

Again, I am not a lawyer.  I have not trained as a lawyer.  This is not legal advice.