LRI or Buffer pH is a measurement of how much "extra acidity" (hydrogen) is attached to the soil cation (nutrient) exchange sites (CEC). A lower LRI number means more sites holding hydrogen. That means more lime (or more accurately, the more carbonate) is going to be needed to neutralize all of that acid producing hydrogen so that pH will rise to a healthier level (pH 6 to 7). Yes, if there is a lot of hydrogen (a low LRI) that is indicative of of a lot of nutrient sites or greater CEC. Greater CEC can mean a heavier soil or a LOT of stable organic mater or a some combination. The lime application will raise the LRI number, it is not going to change the type of soil (clay content).

I don't know which buffer pH test was employed to get the LRI number, so I don't know which chart should be used to determine the total amount of lime needed to raise your soil's pH to 6.5. Experience tells me that it is very likely that more than 100lbs per thousand square feet will be needed. Having said that, don't violate the guidelines. more than 100lbs should not be added in one year in two 50lb applications. One in the Fall (NOW, don't delay) and one in early Spring soon after the ground thaws.

Having the pH between 6 and 7 will make some needed nutrients more available to the grass and lower the availability of other nutrients that could be toxic at low pH. A good thing.

You are correct. Nothing you can do about phosphorous. Don't add more. It isn't toxic to the plant at that level, but it is likely creating some imbalances and your soil will not have the benefits of a nice hyphae to have. (Leave that to Becky if you have questions).

Follow Rutgers recommendations. Do the 50lbs of lime and the potassium (5lbs/k of 21-0-21 for a total of 1#N and K if you can find it). Suggest you aerate the area before adding the lime and K, if it's in your budget.

EDIT: you should get another soil test nex year about this same time, before adding more than the 100lbs of lime they recommended. They will recommend any addition of lime needed and any other nutrient amendments in that new test.