For what its worth, I believe I started off with a 560 credit score. I had collections and judgements on my reports that weren't mine. The one credit card I had with a flimsy limit was closed during the credit crisis and I gave up for a time as it was impossible to fix all this.
About 2 years ago I tried applying for a new home loan (on behalf of my sister). The home was around 50k which is about half my annual income and like you I pay cash for everything and carry no debt so imagine my surprise when I was declined!
I declared war! I read every post I could on forums like this and systematically attached each and every derogatory item on my credit report until it was spotless.
My FICO is now 730. At one point I was as high as 820. I have several great credit cards and limits in what I consider usable levels (10K+). So it will take a little time and some work but you can certainly see great results if you keep after this.
1 - Start with your credit reports. Order all three via snail mail (you can do so for free once a year) and if there's anything bad on them, create a plan of attack for getting rid of those items. Sounds like it should be easy it sounds like any bad items are past the statute of limitations (SOL).
2 - Everyone has a different preferred approach. Some favor store cards, some favor secured cards and others suggest building a relationship with a credit union as they tend to be more flexible on these things. I dont have real advice in this area unfortunately. I dont think a few hard inquiries will hurt at this point. Whatever you get will likely be small but it can be something you can grow.
3 - If your current homes are under a legal business entity, perhaps your business partner can get a business credit card for each of you?? Some report to personal credit reports, others do not. It would help create a separation of business charges from personal charges which is something I imagine your accountant may have advised. Incidentally, if you are incorporated -- I hear annual stockholder meetings in Hawaii are rather nice, and its a legitimate business expense to boot