Welcome to the Q&A Forum
Browse the forum for helpful insights and fresh discussions about all things SEO.
After more than 13 years, and tens of thousands of questions, Moz Q&A closed on 12th December 2024. Whilst we’re not completely removing the content - many posts will still be possible to view - we have locked both new posts and new replies. More details here.
If it's not an urgent issue ... install analytics now and collect data for a month or two. You'll also want to install Search Console. Basically, if the pages have incoming links or traffic you'll obviously want to move them but if not and they're low quality they should probably be left behind.
If your current site is secure and you want to redirect to the new one then yes, you'll need to keep your SSL cert. If you don't you'll get a certificate error and the redirection will hit that snag and perhaps worse, the link weight will hit the same one. Someone linking to the non-https page will redirect but the https will generate the issue. Annoying I know ... now you get to pay for two certificates.
If anyone else knows a way around this I haven't thought of I'd love to hear it but I had to deal with this a few months back and the only solution was to setup the old domain with it's own hosting and certificate and deal with the redirections at that level.
They will recrawl it but equally important is that traffic following links to your site will get where they're supposed to go. A good rule of thumb with search engines as well as humans ... it's never too late to do the right thing. 
(Of course, with a Penguin penalty you might go bankrupt before they get around to rewarding you for good behavior) 
Hope that helps !
Always a good rule of thumb to follow Josh's advice here. If Google gives you an opportunity to give them more info about your business or site ... drink the Kool-Aid and do it.
Provided that there are no links to http://clientname.com/products/ and obviously it's not part of the breadcrumbs you're fine though you may want to consider adding these pages to give you a point to all products (of a list of categories of products), etc.
But no ... the structure itself isn't inherently going to cause problems.
What you're referring to was discussed by John Mueller a while back and Andy's right ... it's devalued.
You can read a bit about what John said or listen to it yourself at https://www.seroundtable.com/google-hidden-content-tabs-19534.html.
Cheers.