Hi vsteele,
Welcome to the Mattress Forum! 
There is more about the many variables that can affect the sleeping temperature of a mattress or sleeping system in post #2 here that can help you choose the types of materials and components that are most likely to keep you in a comfortable temperature range. In very general terms … the materials, layers, and components of a sleeping system that are closer to your skin will have a bigger effect on airflow, moisture transport, and temperature regulation than materials, layers, and components that are further away from your skin and softer mattresses or foam toppers will tend to be more “insulating” and for some people can sleep warmer than firmer versions of the same material.
The Quatro Sleep mattress is one of many boxed bed mattresses manufactured in China, and your supposition about it being compressed for a longer period of time is probably valid. Post #6 here describes in more detail about mattresses imported from Asia or China and which may have been compressed for long periods of time in either shipping or storage before being purchased. Being sourced in China would make something like this somewhat of a risky purchase IMO. Additionally, the Quatro uses 1" of 1.8 lb memory foam, on top of 2" of 1.5 lb polyfoam (or memory foam, depending upon what page of their site you are rading), on top of a 7" polyfoam core of unknown density. The upper two foam layers are lower in density and durability than what I would recommend, and there is no listing for the density of the polyfoam core, so again this would be an item I would provide extreme caution against considering.
There’s a bit of information here about different sleeping styles and here regarding different body types that you may find useful when selecting a mattress.
Memory for does a very good job at minimizing motions transfer (although you’re bound to feel some through your Zinus frame), but it will generally be the least breathable and most insulating of the foam comfort materials, so temperature can be an issue (see my link earlier in this post regarding temperature regulation). If you do choose memory foam you’d want to make sure that you don’t choose something with so much memory foam on top that you sink in too deeply.
While I can certainly help with “how” to choose, it’s not possible to make specific suggestions or recommendations for either a mattress, manufacturers/retailers, or combinations of materials or components because the first “rule” of mattress shopping is to always remember that you are the only one that can feel what you feel on a mattress and there are too many unknowns, variables, and personal preferences involved that are unique to each person to use a formula or for anyone to be able to predict or make a specific suggestion or recommendation about which mattress or combination of materials and components or which type of mattress would be the best “match” for you in terms of “comfort”, firmness, or PPP (Posture and alignment, Pressure relief, and your own Personal preferences) or how a mattress will “feel” to you or compare to another mattress based on specs (either yours or a mattress), sleeping positions, health conditions, or “theory at a distance” that can possibly be more reliable than your own careful testing (hopefully using the testing guidelines in step 4 of the tutorial) or your own personal sleeping experience (see mattress firmness/comfort levels in post #2 here).
I’m not sure what you’ve read since you found the site but just in case you haven’t read it yet, the first place to start your research is the mattress shopping tutorial here which includes all the basic information, steps, and guidelines that can help you make the best possible choice … and perhaps more importantly know how and why to avoid the worst ones.
Two of the most important links in the tutorial that I would especially make sure you’ve read are post #2 here which has more about the different ways to choose a suitable mattress (either locally or online) that is the best “match” for you in terms of “comfort”, firmness, and PPP (Posture and alignment, Pressure relief, and your own Personal preferences) that can help you assess and minimize the risks of making a choice that doesn’t turn out as well as you hoped for and post #13 here which has more about the most important parts of the “value” of a mattress purchase which can help you make more meaningful quality/value comparisons between mattresses in terms of suitability (how well you will sleep), durability (how long you will sleep well), and the overall value of a mattress compared to your other finalists based on all the parts of your personal value equation that are most important to you (including the price of course and the options you have available after a purchase if your choice doesn’t turn out as well as you hoped for).
Outside of PPP (which is the most important part of “value”), the next most important part of the value of a mattress purchase is durability which is all about how long you will sleep well on a mattress. This is the part of your research that you can’t see or “feel” and assessing the durability and useful life of a mattress depends on knowing the specifics of its construction and the type and quality of the materials inside it regardless of the name of the manufacturer on the label or how a mattress feels in a showroom or when it is relatively new so I would always make sure that you find out the information listed here so you can compare the quality of the materials and components to the durability guidelines here to make sure there are no lower quality materials or weak links in a mattress that would be a cause for concern relative to the durability and useful life of a mattress before making any purchase.
Many of the site members feature products using memory foam, and they certainly can be a good place to start your search and learn about such products.
As you are on a more restricted budget, post #4 here and the posts it links to also include many of the better lower budget online options I’m aware of as well.
Posts #1 and #2 in this topic also includes more information about the new "breed’ of “simplified choice” mattresses (aka “one choice fits all” or “universal comfort” or “bed in a box” or “disruptor” or “millennial” mattresses) that are available online as well that are typically in a budget range of between $600 and $1000 (queen size).
Hopefully that information gives you a good start in your search. If you have other questions after reading through this, post back and I’ll do my best to answer them.
Phoenix