I think a simple answer to your question is yes, one can make the same dish in both a flat bottomed pan on an induction cooker and a wok on a gas flame. I might say that it might be easier in the wok, as it was intended for this style of cooking. I might also add that what makes stir frying different from other types of cooking are essentially two things: heat and movement. Stir frying requires high heat and continuous movement of the food to avoid burning at said heat level. As long as both of these requirements are taken care of then I would say you are stir-frying.
Now cooking in a flat bottomed pan will raise some slight difficulties. Depending on how high the sides of the pan are, will make it easier/harder to move the food around. Having high enough heat is another issue when stir-frying. If the sides of the pan are too low, then moving the food will be a challenge, and the food will burn easier or be outside the pan. If the heat is not sufficient, one is not stir-frying but rather steaming the food. But this would also affect an underpowered wok as well.
The use of the induction cooker, with sufficient amounts of wattage or heat will take care of the heat requirement. As I find most induction cookers directly heat the pan enough to fry the food, compared to a regular gas stove,(not professional or home chef professional burner), with anaemic heat output.
As a note, there are now induction cookers that designed for wok usage. I have one, and I find it better than the average gas stove in heat output, but lacking in the actual cooking surface area needed to cook food efficiently and quickly. Smaller batches of food need to be cooked in the wok for it come out right. I also find you need a certain sized wok to fit inside the wok induction cooker for it to work properly. If the wok is too small it will not fully utilise the already small cooking surface and will not be stable either. Too large a wok and it will be stable but the cooking area is also reduced. The ideal wok for my induction cooker is a 36cm wok.
Now onto your second question which is more the logistics of cooking with a (round bottomed) wok on flat cooktop. I would answer this with, yes you can with some modifications. If the range is gas then it's less of an issue, but if the range is electric or induction, I would say it might not be worth the effort. With a gas range, the wok would only need to have a wok ring to make it work. The ring takes care of the stability issue, so long as you have enough BTU's.
With an induction or electric range, the wok ring would provide stability but it might not provide the needed contact to the cooking surface to generate the amount of heat needed to stir-fry. Even with contact, the induction/electric cooktop would only contact the very bottom of the round wok. Not providing enough cooking area for efficient and quick cooking. So it may not be worth the effort.
I think this answers your questions.