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So ideally you would do page to page redirects which would be on the .htaccess level. Meaning sitea.com/topicx redirects to siteb.com/similartopicx If you don't have that ability/don't want to do that, use the DNS 301 and point it all to siteb.com. I would also first verify both websites in GWT and tell Google the website is moving locations.

This is a really tough one to answer because while there are examples of companies doing this in stages and examples of this being done all at once, it's impossible to know what the results would have been if any one example had been done differently. I am tempted to say that the more changes you can make all at once, the better. The reason for this is that if you go through one set of big redirects to go from www.domain.com/url-structure to www.newsite.com/new-url-structure, that's only one set of redirects Google has to process. If you do this in too many stages, Google has to find and process each set of 301s every few months. The more times you 301 a URL, the less authority it seems to retain from the original URL. This is disputed and debated, but it definitely still seems like a poor idea to go through more 301s than you really need to. 301s can definitely result in temporary ranking problems, so the fewer of these you need to endure, the better. That said, this is a huge task with a lot of changes and it doesn't sound like they'll be able to do the whole thing at once?

In general, it is best to do all of the changes (the domain, branding, and URL work) all on a development site (that is blocking search engines) and then release the new site only after everything is complete. If you space it out piecemeal over too long of a time, it will look very messy both to Google and your website visitors. Releasing a new, properly-migrated website all at once is the quickest way to mitigate any "SEO" damage and return to your prior traffic and ranking levels quickly. However, it'd be impossible for me or anyone to be more specific without doing a deep dive into your site. However, there are some general best practices for site migrations. For reference on all of these, I'd take a look at what Moz did when migrating from seomoz.org to moz.com. Moz has a webinar and associated blog post that goes through all of the technical details and results. Aleyda Solis also has a good migration checklist with links to more resources here. I hope that helps -- good luck!