TL;DR: You can use any fingers you want except the thumbs.
You can see the reason for that in the diagram below. Notice that there are two arteries supplying each finger that run down both sides of the finger, but there is a single artery supplying the thumb that runs down the middle of it. That means you can't feel the pulse in your fingers when you press ...
According to Minnesota Dept. of Health:
A normal level of oxygen is usually at least 95% or higher. Some people with chronic lung disease or
sleep apnea can have normal levels of around 90%. The SpO2 reading on a pulse oximeter shows the
percentage of oxygen in someone’s blood.
The instructions that came with my home-use pulse oximeter (in German) roughly ...
As far as I know, the only blood thinner that is known for its potential to reduce platelet count is Heparine. See e.g. Wikipedia, HIT - heparin induced thrombocytopenia, www.en.wikipedia.org/wiki/Heparin-induced_thrombocytopenia
"If someone receiving heparin develops new or worsening thrombosis, or if the platelet count falls, HIT can be confirmed ...
As you correctly mentioned, Korotkoff sounds are caused by a turbulent flow wave, going through the artery - which, by definition, occurs when the heart muscle contracts. So yes, what you hear is the same as what you would feel when taking a pulse.
*Additional note: in some cases peripheral pulse might not represent the exact heart beat (see pulse deficit)