The answer to your first question is yes you may need a Digital TV Converter Box for your current TV. The fact that your TV has a built-in DVD Player does not mean that your TV has a Digital ATSC Digital Tuner built-in. In fact the older your TV is the less chance it has of having a built-in ATSC Digital TV Tuner.
You need to look at your TV's Owner's Manual in the specification section to see if it shows having a built-in ATSC Digital Tuner. Do not confuse this with a NTSC Tuner, an NTSC tuner is an Analog Tuner and will need a Digital TV Converter Box in order to receive Digital TV signals.
If your TV does not have a built-in ATSC Digital Tuner then you need to look at the options below on what you can do to start receiving Digital TV signals.
In searching the internet I have found some Digital TV Converter Boxes that have been modified (hacked) to run on 12 volts. I am not going to link to those boxes as the modifications have been done without the converter box manufacturers permission. I will tell you that the modified boxes I found were expensive running anywhere from $99.00 to $199.00. Of course there are no manufacturer warranties on those boxes.
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