If I get HH do I have to start all over with a new charactor?
No. Any character from before you started the battle for Hoover Dam can play HH at any time.
If your only saves are during Hoover Dam though, then you will have to start a new character. Sorry.
Also is there any problem with HH when your playing hardcoe mode? I have read that there were some issues with this.
Well, there's the fact that bullets eat up a lot of your weight allotment for starting the DLC (75 pounds; 100 if you have pack rat, strong back, high medicine, or high speech and take certain dialog options), but beyond that, I've had no problems.