In a way, you are moving to a new website. You are keeping the products, but basically changing almost everything else: the URLs, the internal linking, the HTML of the pages and the back-end (the cart software). It's impossible to say what the long term consequences might be without knowing the details, but if the new software makes it much easier for you to maintain your site, then that'll give you time to work out any quirks that might come up. This is a big move, so plan it very well and monitor each step to make sure that it works as planned. You will possibly see quite large fluctuations in indexing and ranking, so be sure that you are able to "swallow" them (pick a slow season, have some ads budget available, etc.).
- Changes in content and internal linking will affect your rankings, at least in the short term.
- Don't worry about "too many redirects" for the whole site, but make sure that you only have to do one per URL (don't redirect in multiple steps). Make sure the redirects (old URLs) are not disallowed via robots.txt and test them via Google's Webmaster Tools. Also test all URLs to make sure they redirect (crawl the old site with a tool like Xenu, then check all of those URLs once you have the redirects set up). Yes, this is a bit of work :), but it's better to be certain about this. Keep in mind that as the redirects are being found, your site's structure will be "rediscovered" like a new site would be, so this is definitely something that will affect crawling, indexing and ranking for a while. Set up Sitemap files for old & new URLs and monitor your progress in Webmaster Tools. Be very, very patient.
- Advantages vs disadvantages ... that's something you need to determine yourself. A site move like this is a big move, at the very least you need to plan each step and set expectations properly.
Being able to clearly monitor the progress (via Analytics and indexed URL count) will help you to stay calm during the move. At any rate, try not to do anything drastic should things take a bit longer -- if you did your homework and the technical side was properly planned & tested before the move, then at worst you'll have to wait a bit longer than expected.