Here are a few recommendations for cottages in the Pines neighborhood. I don't recommend staying anywhere south of Pennsylvania Ave, or on the West side of Rehoboth at all. *The cottages I'm listing are all managed by Lingos, because that's who we've always used: #69 Oak, and also #s 35, and 7 Henlopen are lovely cottages with at least four bedrooms and lots of charm. They've been well-maintained and in some cases renovated, but haven't been so "redone" to have lost their uniqueness. I really like 69 Oak for it's huge porches - really more like outdoor living spaces, and because the mattresses seem to be replaced regularly. The kitchen is average-sized buy normal house standards, but huge by older cottage standards, and it has some special custom features, such as a swing on the back porch, and a hanging French door that closes off the kitchen from that area, as well as a spiral stair case going up to the upstairs porches and bedrooms, and to a widow's walk with a small roofdeck for sunning. There's also a hot tub offered as an option but you have to pay a little extra for it because they have to have someone come and set it up and maintain it.
My Grandfather built #40 Henlopen and it's still in the family (owned now by my cousin), however a family usually rents it for all but a few weeks of the season and I'm not sure when it's free. It's a really nice--and very definition of the word "cottage" type of place. Has great front and back yards too.
There are also several very nice homes in Henlopen Acres you should check out. Those are more like year-round homes and might have more of the amenities you're looking for.
Lastly, someone told me that Bethany Beach has rentals and that it's a good alternative to Rehoboth's crowds during the high season, My Aunt used to live down there but I only visited her a few times and didn't really get a feel for the area or the availability of cottage rentals, but a quick search online shows listings under Homeaway.com. Bethany Beach is only about 20 minutes from Rehoboth attractions. The cottages also seem to be very reasonably priced.
I hope this can be helpful to you.