The oven preheats, but then "cools down" before I

Preheat is helpful when you're not cooking for more than a few minutes. If you're going to cook for 20 minutes or more, then most likely you don't need to pre-heat.

To pre-heat, simply turn your toaster-oven on 10 minutes before you put the food in.

If oven is electric pleace check that the heating elements are ok, could be that only one or two elements could be the only heaters hence luck power to maintain temparature when open or gets too slow to preheat.

E08 Error Message on a Samsung Electric Built-in Oven:

The E08 preheating error code appears if the set oven temperature is not reached within 50 minutes. If this code appears, the reason might be due to overheated thermostats, a defective heater or a defective relay; in which case a repair will need to be arranged.
However, if food has been put into the oven during the preheating time that may cause the error code, since the presence of food cools the oven and so it doesn't reach the right temperature. The error message can be removed by turning the oven off and turning it on again.
To prevent this error code from happening again, the oven should not be used during the preheating period. The preheating period ends just after the sound signal and the icon in the display have turned off. When the oven gives the signal, it's close to the set temperature and your food can be put in.

Others are having problems with this unit. Have the manufacturer support their product and fix it. It should be covered if it is new. Don't delay in hounding them. Time will expire and you will be left with an expensive ordeal.

preheating gives u prepare temperature of oven for some specific cooking
Don't leave the oven door open too long when putting the food in because the temperature will drop a little but that is not horrible and shouldn't be a cause of great concern. Each time you open the oven door the temperature will drop a little.

they have a fuse that blows if the cavity gets to hot looks like a small thermostat

Are both of your heating elements lighting up when you are preheating? It sounds like the oven thinks it is pumping power into both elements and one isn't working. Oven elements are generally easy to remove and replace!

You may have already solved this but either your convection motor is not on or you have an air leak. I would get the repair man out since it is probablt still under warranty.

Cold Stove, you may not be able to. The issue here is that oven manufactures put on their units a fast preheat. However it isn't possible to preheat an oven for 7 to 9 minutes and bake short time bake items properly. Also the oven sensor will respond faster than the absorbation rate of the sheet metal in the oven liner. To preheat an oven properly takes 15 to 20 minutes before putting in the food. I see this will a lot of current ovens. Sounds like a normal format of the control.