I've been finding good investment properties in a different state, however, the school is ranked very low! I also found good schools with not so nice homes. How much of an influence does the school have on the renting and/or selling of a property?

In my experience, it is a huge factor.  Like, literally #1.

I think it depends.  While schools can be VERY important if your property is family oriented (3+ bedrooms, large yard, etc), it can be far less important if your house is in a retirement community, is a studio or a 1 bedroom, is a trendy "tiny house", is in a work/live space, is right downtown and close to night-life or is otherwise more geared to those without children or who are "pre-child" or "post-child".

I have many friends who are either empty-nesters or who are middle-aged and childless that just love finding funky houses, close to the stuff they think is cool and who never even know where the nearest school is, much less know or care how they perform.

(Also, insert rant about standardized tests and how school rankings rarely tell a complete story.)

I also feel it is important but I like the other factors that you pointed out. I wanted to get a property with a good school district. However, since I am looking outside of my own state can there be good cities that just happen to have a bad school district. Is this possible or is it assumed that if the school is bad  then the town is not a good area?

I would look for good schools with not so nice houses.  You can always fix the house but not the school problems.  Good schools are just part of what makes an area desirable. 

I feel good schools are a must if you are looking for family oriented properties. That could be small families with just one kid through bigger families with a few kids. All parents ( I assume) want to get their kid(s) in the best school districts and will consider that first and foremost when looking to either rent or buy.

However, if you are looking at more of a cool downtown and hip vibe with young professionals and singles, then schools will not matter.

Example - in NYC, a lot of families want to live in certain parts of Long Island due to the quality of schools. Not only are they willing to pay higher prices for the property, but also the crazy high taxes (a LOT of the tax burden is for the schools). On the other hand, the singles and young folk will pay an arm and a leg to rent or buy in the city or certain parts of Brooklyn because it's fun and cool - they don't care about the school districts. When they have kids, then they'll most likely try to move to LI.