I have constructed the following configuration, as shown in the image below.
$ABC$ is a triangle with the circumcircle and incircle drawn, and point $O$ is the incenter. Furthermore, point $D$ is the point of contact of the incircle with $BC$. $E$ is a point on the circumcircle such that $\angle OEA=90^\circ$.
I can see various properties in this setup, however I am struggling to prove them. How can I prove that line $OE$ is parallel to line $BC$, or equivalently that $\angle EOD=90^\circ$, or even that points $E, A, B, D$ are concyclic.
Any help with this would be appreciated.
I don´t know if you are trying to troll but anyway. Here is a similar configuration that proves your first two conjectures to be false
The third one is obviously false because the circumference through $E,A,B$, which is the excircle of the triangle, doesn´t pass through $D$.