Just about any semi-firm cheese that has been aged for a bit is OK, Olive Oil in a checked bag (liquid) is OK. Do be aware that you need to declare the cheese if asked about dairy product, and despite "regulations" posted, the USDA people have the final say and anything is allowed at their discretion.
Yes, you can get most anything in the US, but if you have room in your luggage, you can get Pecorino especially for a fraction of the price in the US plus many other cheese options that you will not see in the US. For Olive Oil, as others said, look at tins, otherwise heavier bottles. I would go for top end, extra virgin, small batch stuff. Yeah, you can get lots in the US, but not as fresh, unfiltered, and I suppose unpasteurized as what you will get in the there.
Other good food options are high quality dried pasta, canned or jarred anchovies or other seafood options, then shop canned, dried, or packaged items that catch your interest.