ja,
i'm not quite sure how accurate svnightwatch's response is. i'm not an attorney.
if you start a business in Ohio, you can register as a foreign company in the state you want to do business in. however, if you're going to be buying real estate, talk to an attorney/CPA, registering with the state may not be absolutely necessary.
now, if you're going to offer some kind of service like a property management service in another state, then that's different.
as far as " licensing" goes, there really isn't a need for that (again check with an attorney of course). licensing is only needed if you're doing work that requires it and buying real estate, that your company owns, will not require you to have a license to do anything. building permits perhaps and variances and such, depending on what you'll be doing to the property - but you will not need a license per se.
any person or entity that buys real estate - owns it and can do whatever they want with it - (as long as any deed restrictions are followed if any and as long as no code violations result).
so registering in each state with a new company for business you'll be doing there is ABSOLUTELY not necessary.
registering as a foreign limited liability company may be a good thing to do, but best to check with an attorney.
remember KISS
keep it simple silly....
why do you want to live in ohio but do business in another state? how will you manage the investments?