As others have said, $200-$300 is not a lot for a complete set-up. You are going to have to make compromises. And you have several options.
I take it that you have not slept in a hammock, yet? While many people feel that hammocks are even more comfortable than their beds (and have become full-timers), hammocks are not for everyone. Some never manage to get comfortable. Since your trip is a car camping trip, I would suggest that you take your tent and pad in any case, so that you can bail out if you find you cannot sleep in a hammock.
Would you consider a bridge hammock as well? If so, the Eureka Chrysalis might be just what you have been looking for. It should be just about long enough for you, it usually comes with the rainfly included (but you can buy a bug sock separately and use it with a tarp), and it's perfect for pad use. This means you would have no problem using it *comfortably* (in contrast to most gathered end hammocks) with your existing pad. Bridge hammocks also allow for a more traditional sleeping position, which might make the switch easier for a newbie. It would also be within your budget if you shop around a bit.
There are two disadvantages: bridge hammocks are heavier, and the Chrysalis is not the lightest bridge hammock out there (but one of the most comfortable ones). But if you want to use it for hiking later on, you can always shed some weight by switching out the suspension and possibly replace the spreader bars with hiking poles (I haven't tried that, though, and don't know how well that works). The other disadvantage is, that the enclosed system is prone to condensation issues and you don't have the extra coverage to cook etc. But you can always ditch the rainfly later and use a tarp instead. For car camping it should not be a huge problem.
Of those hammocks I only own the DangerBird, but Hammeck hammocks seem to have a fairly similar build. The DangerBird comes with a top cover, though, while the Netty only has the bug net. The RoamingGnome is DH's net-only hammock, and the Hammeck Envy would be comparable to the DangerBird. Both companies make great hammocks and you have many (maybe too many for a newbie) choices.
Personally, I would stay away from the Roo. I don't own one, but I think it is very expensive for what it is (a simple no-net hammock). Since you're on a budget, there are definitely better options. I think Dutch's simple hammocks have already been mentioned. With your preference for a semi-firm mattress I would go with NylonD. Get a double layer to be able to use a pad. Pads are not the best option in gathered end hammocks, but they get the job (of keeping you warm) done. As long as you're not 100% sure that you want to stick with hammocks, I would not shell out for an high-end underquilt. Dutch also sells a bug net. For hammock, suspension and bug net you would need to spend about $150. That leaves $50-$150 for a tarp.
For $150 you can get a great silnylon tarp - the WB Superfly is a 4-season tarp with great coverage for pretty little weight. Many use it year round. The WB Edge is smaller, but would be perfectly fine for car camping if you don't expect week-long rainstorms, and it's a great summer or 3-season tarp. The WB Mamajamba is in the middle. If you don't plan on hanging in deep winter, it would be a perfect all-round tarp. Although I own the Superfly and the Edge, the Mamajamba is my most-used tarp.
If you wanted to spend less, you could also go with a tarp like the Kelty Noah. You probably don't want to take it on extended hikes, but many people here use it for car camping or kayaking/canoeing. It's pretty cheap and offers a lot of protection.
There are many more hammock and tarp makers, but I guess you already know that If you stay with hammocks, you'll likely start a collection anyway. There are very few people who only own one hammock... So don't sweat it and simply pick one that appeals to you. It's easy to sell high-quality gear here and try something else.
It seems that most people use whoopies, because they're lighter than a webbing suspension. However, I find webbing/cinch buckle suspensions more flexible (no problem with very narrow hangs) and easier to use. I started with the WB webbing suspension and switched to whoopies for a year or so, only to switch back to webbing. But it's a matter of taste.
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