Dry ice is a bad idea (bottle will explode and it wont take several minutes, if you do it right it will take seconds, also it’s extremely sensitive to shock.)
As for the mentos aspect my friends and I have been doing it since at least last year. Only we've added our own little flair here and there.
We started off with coke and mint mentos and a paper launching tube. We then started experimenting. we've all done several papers on our results (easy A's) we used several different types of soda (Diet Pepsi, Pepsi, Diet coke, coke, cherry coke, vanilla coke, cherry Pepsi, mug, A&W root beer, ginger ale, store branded cola's, grape soda's orange soda's, etc) we've done separate experiments to see what the perfect number of mentos to drop in is (starting at 1 and going up to 30) we found that diet coke along with 11-14 mentos(one pack) is the best combination.
Other interesting things we've found is that diet soda's work better across the board compared with there regular sugared counterparts, this is because the sugar suspended in the solution restricts the CO2 from coming out of solution. Also diet sodas are pumped with more CO2 at the factory
Warm soda works better than cold or frozen soda though we cannot explain why this happens.
For anyone curious as to why this reaction takes place mentos contain a gelatinizing ingredient called Gum Arabic which reduces the surface tension of liquids. Also mentos have a chalky exterior which dissolves as they fall leaving millions of tiny pits on the falling mentos. These pits act as nucleation sites for CO2 bubbles to form on.
as for out modifications to this reaction: when we first started I wanted more height so I took a cylindrical aluminum blank and machined it into a nozzle that simply screwed on in place of the cap, we then took the mentos and drilled holes thin the center (most of them will split when you do this). The ones with holes in the center were then strung up to a piece of fishing line and suspended above the liquid surface. With the nozzle in place we'd drop these in and we'd get a serious fountain, usually around 20-25 feet.
Next we improved our launching mechanism, at first we were just using a rolled up piece of paper but we soon moved to a piece of vinyl tubing. Soon after that we went down to the science lab and grabbed some test tubes (these work the best). since we had to give the test tubes back we went looking for another source, we found them in a cigar shop, turns out some cigars come in glass tubes, we just asked the owner to save us a few and he did.
After a while you get bored of this, so we moved onto the next best thing, how do we make this reaction destructive? The answer was simple, get one person to hold the bottle, another to drop the mentos and as it started to explode have a third person standing by to screw on the cap to this exploding bottle. the reaction will pressurize the bottle so that cap can barely maintain the pressure (moving the bottle fast, dropping it, or squeezing it the smallest amount would result in the cap blowing off violently and soda spraying everywhere. we got this reaction down and we would throw the bottles, when they hit the ground the cap would blow and the pressure would propel them through the air like a rocket. you can also lay them on the ground and have some one run and kick it but we've had a few done that way backfire (2 liters bottle Vs. Crotch + pain)
Just a few things to consider if you’re going to try this