@Lou Metore Happy to answer these questions. Let's start with what happens when you leave Investor Carrot or Lead Propeller.
Investor Carrot TOS:
"Upon cancellation, all of your content may be immediately and automatically be deleted from the Service. This information can not be recovered once your account is cancelled."
Lead Propeller TOS:
"Eligible Licensees. This Software is available for license solely to SOFTWARE owners, with no right of duplication or further distribution, licensing, or sub-licensing. IF YOU DO NOT OWN THE SOFTWARE, THEN DO NOT DOWNLOAD, INSTALL, COPY OR USE THE SOFTWARE."
and
"Reverse Engineering. You agree that you will not attempt, and if you are a corporation, you will use your best efforts to prevent your employees and contractors from attempting to reverse compile, modify, translate or disassemble the Software in whole or in part. Any failure to comply with the above or any other terms and conditions contained herein will result in the automatic termination of this license and the reversion of the rights granted hereunder to FREEDOM DRIVEN LLC."
So, as is fair, you can't sign up for either service, then take the content you were leasing, and set up your own custom page with the same content. I'm not an attorney, but in the case of Investor Carrot, if you customized all of your content used on IC, I'd hope you would be able to take your custom content with you when you walk, just not their template content that comes out of the box. Sounds like the same goes for Lead Propeller, except that 'reverse engineering' clause, I'd want to better understand what is covered in reverse engineering, and I'll explain why.
Any SEO value given to a page, whether it be a homepage.com or a homepage.com/about-us, is organized based on the url. So, if your Investor Carrot website's "about us" page was homepage.com/our-company and you move to your own custom website, you will need to maintain the url structure, and be sure that your new page is also exactly homepage.com/our-company (not my-company, or about-us, or learn-more, as those will not maintain the SEO value of the original page).
It's kind of like redesigning a kitchen, Google knows the kitchen room of the house as 'kitchen' and even if you gut the kitchen and redesign it (change hosting, and web design), as long as you don't rename it as the "/bathroom" or the "/bedroom" or EVEN the "/kitchen-area" it will still recognize the page and maintain the prior SEO value. It must be exact. This is one of the biggest challenges with site migrations, is maintaining url structure (gets really messy when you're switching content management systems, but if you're going from wordpress to wordpress, it's very simple). Many companies change their web design, and their traffic tanks - because all of their urls have changed, meaning all of those pages (more or less) set the SEO clock back down to zero. You can 301 redirect old pages to new pages, if maintaining naming conventions is not possible, but many don't know this.
Since the homepage won't change, the SEO value would be maintained (and change only based on modifications to content, and meta tags, which you can easily replicate without copyright infringement)
So, as long as IC and LP don't claim 'url structure' as copyright (it seems unlikely, but I just don't know, and they can update their TOS at anytime), then the migration shouldn't cause much impact on your site if your custom site maintains the url structure - but I would venture to say most folks using IC probably don't know the SEO implications of changing url structure, so they're unlikely to maintain it in a site migration. They probably leave, their SEO disappears, and they run back to IC - all because they didn't match the url structure. Again, there's more factors involved, but this is the largest issues when migrating a site.
Mobile: This comes with selecting the proper theme. The theme I mentioned above, Agent Press is mobile responsive, so mobile is not an issue you will have to worry about, assuming that you check the mobile version of the site (go through the entire funnel from homepage to filling out the form) and make sure there aren't any glitches overlooked by the dev. Visit the Agent Press demo page (linked above in this paragraph on your mobile device to get an idea for how the theme responds on mobile). Unlike in the past, mobile is baked into the web design, you don't need to think of it as a separate entity.
You asked about how to choose your designer. I would recommend looking at Upwork, and finding someone who's done a ton of work, has a lot of positive reviews, and be very clear about what you want. Look at who in your area is spending a fortune on Adwords - if they are spending a lot on Adwords regularly, then you can safely assume they aren't just wasting their money, and their website must be converting well for them. Homevestors is a good place to look, they spend a fortune on PPC. http://webuyuglyhousesfast.com/houston/we-buy-ugly... Here is one of their landing pages they're using for Adwords. Pick a website that you want to look like, without completely copying, as a guide for the designer. Make sure you have all of the elements (testimonials, call-to-actions, BBB ratings, whatever you see there). The beautiful thing about online marketing is the transparency in what your competitors are doing. But again, don't copy, but borrow ideas and make them your own.
@Lou Metore If you find a designer on Upwork with great reviews, and you're ready to pull the trigger with him/her, I'd be happy to jump on a quick call with them and let you know if I think they're legit.
As for getting the website ranking, you can hire someone for a few hundred dollars to modify keyword density of your pages, and update meta tags, but again, my biggest focus would be paid advertising (very targeted, start with [sell my house fast] limited to a 10 mile radius from your business). It may only cost you a few hundred dollars per month (depending on where you live, and the volume for that one single term), but the lead quality is going to be very high.
The value of onsite SEO in a competitive market is almost zero. You're going to need off site SEO (link building) to compete on terms WORTH competing on. I'm generalizing, so there are exceptions, but not many. If you DO rank #1 for SEO, you are halfway down the page, under a bunch of PPC ads - So you can worry about SEO each month, or just pay the $15/click (or whatever it is in your area) for HIGHLY targeted terms that indicate high value leads. You can also pick off your competitors by advertising on their terms (but not using their trademark in your ad). Those are a bit more complicated when it comes to quality score, and cost per click, but jus keep it simple at first.
Again, SEO is like building your house on sand, and PPC is building it on a rock (or at least a very hard clay ;) - Any SEO update could come by and completely knock you off your SEO horse, PPC is much less volatile. Money I spent 3 years ago on PPC, still educates my campaign today, the more you spend, the more you learn about your market. There's far less unknowns with PPC.
Think about it, would you buy a company that generated 100% of it's revenue from SEO, or 100% of it's revenue from PPC? The day after closing on that company, an algorithm update could kill all of that traffic - with PPC, you could get banned from Adwords, but you really have to try hard for that to happen (or be in a spammy industry like tech support).