Some operate almost exclusively as a business (like hotels) and others are much more passive.

Our home insurances were done through Geico (as I am sure many of you do) however when I look at their personal umbrella liability insurance policy I can see that (among the 41 exclusion items :) ), some exclusions are quite concerning and actually might not cover us under some of the more common causes of liability lawsuits/loses that we could imagine happening while we rent our rental properties.I wonder if these exclusions typical for umbrella insurance policies OR its just Geico that is not fitting for investors ?

If it was previously used as a rental, you will have to pro rate the exclusion - so you're still out of luck.These are questions you need to run by a CPA who knows your situation.

It's just an exclusion from coverage.  

Do not sign any exclusive agreements and ask for comps when they agree to submit your offer.Sooner or later you'll find who you want to work with.

For example,If I put a house under contract, should I seek to get an exclusive right to market the house?

You being the buyer is important for the purpose of you taking the IRS section 121 exclusion on capital gains should you ever sell.

Once this is accomplished then it becomes sec 121 property and the same primary residence tax free exclusion is available to you on this property as with any property with two additional requirements.  

Not sure how that is handled, if it's still considered your primary for the 121 primary exclusion, but I don't think so.

Can the house be put under non-exclusive contract for the wholesale on a 30-day buffer with the option giving the homeowner to still work their other options?