LPs are difficult to sell on eBay for a couple reasons- one is size, which makes them expensive to mail (although you can use media mail rates), and the other is condition. Condition can be so subjective, and buyers can easily be disappointed once they receive the item. As a result, for the most part many vinyl albums go unsold on eBay. You mention rarity, which is key. No one wants another copy of a Styx album. In addition, the cost of producing CDs is so low now that fewer albums are available on LP only, which has affected the overall market price for vinyl. The good news, as you have noted, is the abundance of computer-related turntables, which make vinyl attractive again. For a long time, the only practical new turntables were high-end deejay models.

I have found 45s are much easier to sell because they are easier to ship (they fit within two pieces of cardboard in a 8-1/2 x 11 envelope), so the cost of shipping doesn't scare away bidders.

EPs are generally more scarce; I would think there would be a good market for those.

Picture sleeves are generally good to sell, especially for 1960s and earlier. I have found eBay to be good for that type of sale.

As an aside, 78s are generally too fragile, and amateur shippers find themselves refunding the price for damaged goods. 78s have to be separately packed away from their covers, and cannot be allowed to shift during transit. They packaging has to absorb every drop or jarring of the package. I'll bet it's easier to ship eggs.

I agree with finding local record stores, even if it means going to the next major city. If your records are really valuable, it will be worth the visit. If they are not, they're probably not worth the listing fees. An honest conversation with the owner/buyer at a record store might tell you which ones are worth auctioning (honesty isn't always available, of course). I would say, in general, that any album they offer at least $5 is possibly worth auctioning. They will generally look through your albums and set up stacks - not wanted, $1, $2, $3, $4, $5, higher.