I have allergies to a lot of skin/hair products. I also am a carry on gal. I use the Eucerin SPF 30 and decant into a 3 oz size container. That is enough for my face and hands for a month's trip. IF I were having to cover other body parts I would be able to purchase in a pharmacy in Italy or France as I'm not as sensitive there. I also buy LaRochePosay tinted SPF moisturizer once I get to France where it is a lot cheaper.
Yes, Italy and France go by the 3-1-1 rule. Use a quart ziplock bag, not a plastic pouch that is marked "TSA compliant" because many of those aren't. IF you want to use one of the plastic pouches, put your liquids IN a ziplock and then into the pouch. The heavy plastic pouches sit up better on a hotel counter so I think that is why many use them. Some airports have installed new equipment meaning you do not have to take out your liquids. I err on the side of caution and have everything set to be pulled out and if not then that's easier for me.
The only thing I've ever had happen at a France airport (CDG in Paris) is when I pulled out my quart ziplock and showed it to the security guy he nodded and waved me thru.
The TSA page says mascara is considered a liquid but I am sure half the population who travels with it has gotten thru security with it accidentally stuck in their personal items. Since it's listed as a liquid, I'm sure the same would be considered for eye and brow liners unless you use a pencil or powder for those.
https://www.tsa.gov/travel/security-screening/whatcanibring/items/mascara
I buy lotion in Europe or use what is in the hotel bathroom's free selection.