Hi alloftheabove,
Just in case you haven’t read it … the first place I would start is post #1 here which has the basic information, steps and guidelines you will need to make good choices.
The size of a mattress is really a matter of preference and comfort but a good guideline is that it should be about 8" or more longer than your height to allow room for a pillow and to keep your feet from hanging off the end and about 16" or more wider than your shoulder width to give you the room you need to sleep without sleeping too close to the edge and to keep limbs from hanging off the mattress. If you are 5’ 8" then this would indicate a mattress that was 76" long or longer and since a twin size is 75" long you are right in the range where it may just be OK. The width of a twin size (38" to 39") provides you with the same room as a couple would have on each half of a king size. If you want some extra length then twins also come in XL sizes which are 80" long (the same length as a king or queen). If you need more width than a twin, then a full size is 15" - 16" wider (53" - 54" wide) and they also come in a 75" length or you may also sometimes find them in a full XL size which is also 80" long. I would also keep in mind though that the cost of a full XL is usually very close to the cost of a queen size (which is another 6" wider at 60") so if you bedroom has the room then it would make sense in terms of “value” to get a queen instead of a full XL.
The size you choose will have a significant effect on the price of the mattress but in the larger sizes then $500 is on the low end of the budget range … especially when you add a foundation as well … and most premium comfort materials won’t be available in this budget range. To move into the range where at least the top layer is a premium foam (such as latex or high density memory foam) you would probably need to raise your budget a little.
Some of the Ikea mattresses use good quality materials and can be good quality and value options (see post #3 here).
Yes … all the Dixie Foam mattresses use high quality materials. Both the Easy Rest and the Dream Bed use high quality polyfoam but of course the Dream bed is higher quality/density than the Easy Rest and I would lean in this direction (I believe it uses 2.8 lb HR polyfoam which is a very high quality material).
Both Dixie Foam and Brooklyn Bedding are very good options and if you end up choosing between these two you would have eliminated any “bad” choices and be making final choices between “good and good” (see post #2 here) which is the most important goal of this site 
Phoenix