Do you flush paper in Greece toilet

Typically, no. Bins are provided next to the toilet for paper products. If you don't see a bin it has most likely been determined that a bin isn't necessary, but that situation is few and far between.

For the most part, no.
Our very nice vacation rental came with an extra outdoor trash can with a heavy duty bag/lid. I don't really remember much about it otherwise. Bathrooms elsewhere were well maintained (other than the occasional public one--surprisingly, no worse than anywhere else).
It is worth it for all that Greece offers, really. But I wish it was like I experienced in Brazil--bidet in the form of hoses were always featured. Perhaps that is not common in Greece because of water consideration (just a random guess).

It has to do with the plumbing. It is very narrow. You do get used to it.

Nope.Old pipe/ sewer infrastructure .Too narrow. Use the can provided next to the toilet.
Agree,you get used yo ot. No big deal.

It is worth mentioning though that it is not universal across Greece. Larger modern hotels may have internal systems (like big blenders, if you really want to know) and better connections to upgraded sewer lines, and even some houses with upgraded septic systems of their own, or new developments, may not require you to "bin-it".

But is is fair to say that tossing paper in the bin is more common than not, and among the Greeks more of a habit or expectation, even if it is not required.

Generally, Greece is no flush - use the trash can. We prefer to use baby wipes instead of TP. Wipes normally locally available, but take some from home to use until you can shop.

We are in Greece now and it’s no toilet paper in the toilets. Not very reflexive for me but am getting used to it. I’ve messed up a few times though!