It's important to remember that composite-to-HDMI converters like these...
... are NOT upscalers or upconverters. They merely convert the 480i composite signal to 480i HDMI. The result will look no better than connecting composite directly to your TV.
If you're using an HDTV, the final result will depend on the quality of your TV's internal upscaler. Most TV's give somewhat mediocre results, with cropping and other issues you mentioned. There's a chance you can go to your TV's picture setting menus to fix that.
Real upscalers are kinda expensive, but incredibly worth it. They specialize in taking the source signal and scale it up to 1080p so your TV doesn't have to.
For a clear picture with no lag, this is the one you want.
http://www.solarisjapan.com/xrgb-mini-framemeister-compact-up-scaler-unit/?setCurrencyId=1