1) My website does not work fully in Netscape; will this affect my listing on DMOZ?
It may or may not. In theory it does not, but if the editor who reviews your site is using Netscape when he reviews it, then yes, it could affect your listing.
2) How long does it take to get your listing on Google directory once you have been successful?
That depends on luck I guess. If Google updates just the day after we list your site, then the answer is one day.
However, if they do it a month later, the answer would be one month. I believe they update about once a month now, but we have no control over when they do it.