Just like the 2 suggestions, it's really best to keep both sites. Move forward with the new site and re-brand your company but don't scrap the original website. Add a banner, write up, some news explaining the re-brand and point the original site to the new site. You'll create a backlink and if the person is genuinely interested in what you have to offer, they'll follow to the new website.

301 any page that are relevant from the original site to the new site. If both sites have a products page then you can point it over. Google will naturally figure out what you did and start passing any relevant benefits to the new site.

Sounds like something built in early 2000s. Tables won't hurt your SEO directly but it will have a negative impact on pagespeed/load and content optimization. Tables use a lot of extra code that makes it sluggish. I generally always recommend a re-program into DIV/CSS. It's faster, optimized, you will see an SEO benefit.

If your client really doesn't want to do the recode, what I would do is setup all the tags, clean the menu, add any rich snippets that apply, then let them know that's the farthest they'll be able to go without upgrading into DIV/CSS.

There might be a link or something directing the crawlers to your site's IP address instead of the original domain. There is potential for getting flagged with duplicate content but I feel it's fairly unlikely. You do want to fix this though, it would hamper your backlink efforts. These steps will correct this issue.

1. Setup canonical tags on all your pages. This lets Google know that 1 url should be linked for this page whether they're on the IP or domain.

2. Set your host up so that anything that directs to the IP is automatically redirected to the domain. This can be done with your hosting company, or through .htaccess, or through PHP. I suggest you do it with the hosting company.

3. Check through your site and make sure no links point to the IP domain. If there are no links pointing to the IP, the crawler shouldn't follow.

It's really best not to change your URL structure around. If you really need to, then definitely make sure you have 301 directs all pointed from the old links to the new ones.

The permalink keywords in the middle don't really apply as much weight as they used to. Using /home-rentals/ and /rentals/ won't immediately relate the pages to those keywords anymore. So with that, set your structure based on the different sections of your site so they don't conflict rather than inserting keywords. So example: "domain.com/search/california/" doesn't conflict with "domain.com/category/california/"

I need to see your pages to give you a better response on the last question. With permalinks, it's always good to match your page title with the page's main keyword. So if the title is 123 Street Ave then the link should be /slug/123-street-ave/. The slug is whatever descriptive keyword for that type of post is. It would be /search/, /category/, or no slug at all.

That doesn't answer your question for the SEO decline though. Chances are you've been affected by the recent Penguin 2.0 update. I'd start by checking my links and seeing if any of those sites got hit. Also check your webmaster tools and see if any notices have popped up.

There shouldn't be any 301 links in a sitemap. A sitemap should only contain links to active pages. So in your case, you should remove all the 301 links and replace them with the new links.

Couple notes - Having 301 links in your sitemap won't hurt your site or SEO unless the sitemap is so huge that you need to split it up into multiple files. But you should really only have the final links in the sitemap, neither people nor bots want to be redirected around. If you properly 301'd the crawlers will automatically update their links.

Changing links around in the sitemap generally won't hurt your site. Especially if the links no longer exist and you're improving the list. There are very few cases where making changes will hurt the site.