If the DVD player is connected to the HDMI port, then you select one of those from the input options. Or are you talking about your cable box connection? How the FIOS box is connected does not matter here. I don't see a port list for that TV, and you did not list the DVD player model and what types of connections it has. Since you have a new TV, may be a good idea to get a DVD/Bluray player with HDMI to use with it.
How you connect your Fios box doesn't affect the DVD/VCR.
AV is the same as composite. The issue may be in the DVD/VCR. The composite out may only work with the VCR. The DVD may work with HDMI and composite. Check the owners manual.