Does my LG 32LD350 TV have a digital converter?

There's no reason. The two things are not connected. Make sure you have the TV on the digital tuner, and not the analogue one.

You can check this service manual for this TV.
There is information on page 76 about LOCK settings.

If you forget your password, press '0', '3','2', '5' on the remote control handset.

service manual

Is it repairable, well, this question can be answered only after opening the back cover and evaluating the damage. From you side, please connect the TV to a different power outlet where other similar devices are working and see. Then try changing the input device to something that sure works. If it does not work even then, this may need attention by a qualified service technician.

Code 0011 worked on my LG 42in LCD.
Thanks to Priya darshan for the codes.
BTW, code 0001 only partially worked ("Mute" did not work).
I phoned Tivo help desk (I am in New Zealand) and they confirmed that LG is still not listed as one of the brands of TV from the list of choices.
Simply choose any of the TV's listed and start entering the 4 digit codes on this site (or others found via Google) - one will work.

Hello
It has been permanantly damaged, and is not repairable. To get it work perfectly, you have to replace the screen. OK.

By looking at this for you it seems to just come off once the screws are taken out. Please double check all screws are out. Additionally the manual may be of some extra assistance, available at: http://www.lg.com/uk/common/product/support.jsp?localeCd=uk&productId=32LD350&uri=/uk/tv-audio-video/televisions/LG-lcd-tv-32LD350.jsp#
Thumps Up!

When you are auto tuning the system will not store a channel which is week by its standard ( at least 60% signal strength ) This done to avoid grainy picture and hissy sound.
You can go into manual tuning and store it.

LG-32LD350-CB.Change audio sistem DK,M,I for B/G.??

Most unlikely. But you can with HDMI cables. The usbs are most used for updating your LCD tv with a usb drive.

Just go to LG website and download the user manual.
http://www.lg.com/us/tv-audio-video/televisions/LG-lcd-tv-32LD350.jsp

http://www.fixya.com/support/r5093881-lcd_flat_panel_tv_troubleshooting_guide
http://www.fixya.com/support/r6150077-basic_lcd_monitors_troubleshooting
Failed TV and Monitors: http://s807.photobucket.com/home/budm/allalbums

Learn about bad caps: http://en.wikipedia.org/wiki/Capacitor_plague