Wiring question from EMT to opposite side of wall
Keep in mind that everything has to be rated for the location;  the boxes, fittings, conduit, wire, all have to be rated for the location in which they are being used.  The question doesn't say this is an outside or wet location, so this answer presumes it's a dry location.
To connect EMT to a box through a knockout, you use a connector.  If the conduit is attached snug to the wall with one-hole straps

you have a small challenge - the knockout in the box won't be quite snug to the wall.  If you have the tool and are adept bending conduit, you can bend a small zig and zag into the conduit so it enters the box square, but that's probably not the case.  So you'll want an offset connector, also called a box offset, which has that zig zag built in, so things mate up square.

If you use hangers to attach the EMT to the wall

they do not hold the conduit tight to the wall, and will not require an offset - things will meet square with a plain straight connector.

Hangers do make things simple, but they cost more than one hole straps, the attachment is not quite as strong, and they have sharp edges and bits that tend to catch on things, and one hole straps look a little nicer.
The NM cable that you're going to fish in the wall must be clamped where it enters the boxes.  You can use a fitting like this one to clamp NM cable in a knockout in the back of the surface mount box:

You'll want to drill the wall a little bigger than the outside diameter of that button so the box sets flush on the wall.