Your shirts definitely need to be collared. “Business casual” does not include T-shirts. Ever.
If you can wear jeans (which is unusual for a business casual environment), you can absolutely wear a pair of slacks. Black loafers should be fine.
As to rolling up your sleeves, look around and see what others do. I would think you’d be alright. Unless you have any unfortunate tattoos. You know, proclaiming your lifelong loyalty to the Aryan Brotherhood, or something like that.
Textured white shirts? I don’t know exactly what you mean. White-on-white dress shirts would be acceptable (if a bit flashy).
It’s going to be difficult to predict what an employer who permits jeans as part of business casual is going to say about untucked shirts and rolled sleeves. This is a bit like saying you’re required to follow Catholic doctrine, but not the whole Pope bit, and wondering if you’re allowed to take birth control.
Your best bet is to observe and modify.
But from a purely aesthetic perspective, my take is this: Definitely tuck your shirt. Rolled or unrolled sleeves is fine. Consider getting something other than black loafers to pair with your jeans. Brown would be more versatile (ideally ones without 90’s square toes that are still inexplicably popular).
Brown loafers look good with jeans. I think black loafers look OK too. Jeans that you wear in the office shouldn’t be faded or torn, though.
Richard Parker is right about those square-toed loafers. Get yourself a pair of plain old penny loafers. You can’t go wrong with Bass Weejuns or Sebago loafers. They were fine in 1965, and they’ll be fine in 2025. And they look good with anything short of a tuxedo.
Any shirt you wear to the office should be tucked in. And, depending on your office, a polo shirt might be a bit too casual for business casual. Or it might not. Look around. When in Rome. . .
French cuffs are not casual, and should only be worn with a suit and tie. Shirts with tiny buttons on the collar? Do you mean button-down shirts, like this? In my opinion, the button-down is the best all-around shirt you can get. It works with a suit and tie, it works with slacks and a sport coat, it works with khakis, it works with jeans. You can’t go wrong. Blue and white are the most versatile colors.
Until you get a feel for what’s OK and not OK in the office, the jeans, brown loafers, and button-down like the example in Saintly’s post are perfect. If you want to be extra caution, just do slacks instead of jeans with the rest the same. Personally, I hate polo shirts (just ugly t-shirts with a collar sewn in, IMO, puke), so you get my vote to nix them completely from your wardrobe, but that’s just me. I’d vastly prefer any shirt resembling a button-down, sleeves rolled up is fine, and even untucked isn’t awful if the hem is cut for it (straight across) - but start with tucking shirts when you’re new!
If you want to build up some options without spending a ton of money ahead of time, hit up your local Goodwill or other thrift stores and see if you can find some dress shirts or the dreaded short-sleeved button-down shirts (the Dilbert look), if you’re seriously wanting short sleeves.
First day or two, wear slacks, a long sleeved buttoned shirt and brown loafers. If you’re too formal, then roll up the sleeves. You’ll soon see if you’re being a bit too dressed up compared with jeans wearing polo shirt sporting colleagues, or you’re fitting right in.
Given that jeans are okay, then it’s pretty likely business casual does not mean blue blazer, loafers and button down shirt.
My rule of thumb, it’s a lot easier to lose the coat and tie, and roll up the sleeves than to try teleport them into the office. Being overdressed is better than underdressed your first day, and I would err on being a bit more conservative. If everyone is wearing jeans, then day 2 wear jeans instead of slacks.