Assistance with cable box set up

I am attempting to set up my new cable box and wireless gateway. I totally understand the procedure for the spitter and the two coax cables.

Here is where my problem is. I am replacing an archaic cable box. The coax cable plugged directly into the cable box. The HDMI cable connected to the cable box from the TV.

The new cable box does not have a male threaded post for the coax cable. After the splitter, do I connect the coax directly to the rear of the TV? 

I would assume that the HDMI still goes from the TV to the cable box?

And lastly, does my DVD player HDMI still go to the TV or does it go to the new cable box in imput?