What is the inclination of Mercury's orbit w.r.t. the Sun's equatorial plane? Additionally, what is the orientation of Mercury's orbit (long axis) w.r.t. the Milky Way?
What is the inclination of Mercury's orbit w.r.t. the Sun's equatorial plane?
From this page, we can see that it is 3.38°.
Additionally, what is the orientation of Mercury's orbit (long axis) w.r.t. the Milky Way?
This is a bit harder to estimate precisely, but we can refer to this view from the IRAS satellite to get a rough figure of 63° between the ecliptic and the plane of the Milky Way (for more information on how to calculate that, check out Aitoff projection). Now just add/subtract Mercury's 7.01° of inclination to ecliptic and you have your result.
EDIT: Because the angles of 63 and 7 degrees may not be along the same axis, there are ways to be a bit more precise: you can take Mercury's coordinates in the ecliptic coordinates system, and transform them into galactic coordinates. If you do that with only 3 different points on Mercury's orbit, you can find the angle made by the plane defined by those 3 points and the horizontal galactic plane.
You can use this website to get your 3 sets of ecliptic coordinates for Mercury, and that website to transform them into galactic coordinates. You can transform those into regular Cartesian coordinates since the website that gave you the ecliptic coordinates also gives you the distance from Mercury to the Sun! Once you get there, it's easy to get the angle made by that plane and the horizontal: good luck!