Room temperature can definitely play a part, but that's only part of the issue and usually only a factor if your room is staying above say 85f. Running it without the case will give you some help, but overall it's a very poor choice because your room isn't going to give proper airflow over the interior parts. On a couple friends old machines I have installed 120mm fans and such to the exterior forcing air in to really maximize the airflow. This I did only because they wanted it. To me it's not a good choice unless you mod the interior fans for the change in speed or match them which is fine. Fan speed needs to be relatively close or you in essence overclock the stock parts which kills the xbox. Plenty of vids on YT about adding USB PC fan Mods if you wanna go that route and it'll cost you maybe $20 in parts.
The addon intercoolers are pieces of shit and should be avoided at all cost.
A good friend of mine let me mod his interior dual fan and it hasn't given him an issue since.
Here is a YT vid with the jist of it. Basically, you're unlocking a dumbass design flaw from MS.
The other tips are all about positioning and environment. Laying it down or standing it up makes no real difference, but I myself always keep it standing as it gives it the max amount of ventilation. Testing has shown though that it makes no real difference, so do it how you prefer. Never sit it on carpet, cardboard, etc as they don't disperse heat. A nice wooden surface is ideal and give it plenty of room to breath (I always suggest 6 inches around every exposed side). This is why entertainment cabinets and such are shit as they tend to provide poor ventilation. Mine is white and has sat on my tv stand (flat top wooden surface that my hdtv, x360, and dvr sit side by side on) for a couple years now and has had no issues. My room stays in mid-70's the majority of the time. I also dust my out casing with compressed air every couple weeks, but that is because this room gets dusty as fuck for some reason.
If you're gonna use a room fan, then keep it at least a foot away from it. They are a bandaid and tend to drag shit out of your air across your machine. Make sure it is facing away to create air drag across your machine.
My business is building/maintaining PC's, but over the years since PS1 I've done everything from modchip installs to case mods. I don't know everything, but I can usually help you figure out how to jury rig something if cash is an issue.