Just so you know where this is coming from, I'm not a mech eng in the field right now, I'm in my 4th year of an ME degree, and have done a variety of internship placements, one of which was at an ID studio (and I'm looking at getting another soon). But based on that internship...
I think there are two main things from ME that are useful to have a sense or knowledge of in an ID setting, manufacturing and structural integrity.Manufacturing:
I've seen designers who don't have knowledge in manufacturing, or are uncomfortable with the knowledge they have do one of two things. They'll either go ahead with whatever concept they feel like, whether it be manufacturable or not, or they'll play it safe and only work within the processes they know. I honestly prefer the former, its better to get the full creativity out and then solve the problem of making it work, but neither is ideal. The solution to me would be to read Making It front to back, and keep up to date with new processes. Also don't be afraid to invent new processes based on existing technologies.Structural Integrity:
Essentially, you don't want it to break, and you don't want it changed drastically when an ME says that it'll break. The reaction tends to be similar to that of manufacturing, they either make the parts way too thick (this is done all the time in engineering as well btw) or way to thin. Either way, the pure design intent isn't being carried through. This is where the kinds of books I mentioned earlier come into play, and knowledge of stress-strain curves, safety factors, ductile vs brittle failure, etc. is important. Ultimately, the amount of time you would need to put in to be able to effectively calculate failures may not be worth it, but a quick calculation for a ballpark stress value never hurt anyone, and would save quite a bit of time further down the chain.
In addition to these, some knowledge of physics and materials may in fact help during the concept stage. For example, most people know that gyroscopes make things stay upright, but most people don't know why (it gets into quite a bit of calculus and matrix algebra). As a quick explanation, gyroscopes turn in a direction perpendicular to the direction a torque (force) is applied, eliminating the positive feedback loop that normal toppling objects experience (which is why tops start turning in circles before they fall). My mind is still racing to find cool product applications for this property. Similarly, some knowledge of materials engineering might help if you can't find the perfect material that you can see in your mind, just spec a certain polymer blend, annealing time, etc. Knowledge of fluid dynamics will help if you have to deal with air or water flow within the device.
I wouldn't consider calculus as terribly important for designers to know for its own sake, if you took it in high school you should have a basic understanding of derivatives, but integrals are more important along gradients such as dam pressure, etc. (the pressure at the top of the dam is 0, the pressure at the bottom is high, what's the total force?). Physics is definitely important, chemistry I would only consider as important as a prerequisite to materials engineering, when you start getting to the molecular level. I don't think at the level you would be looking at them you need to worry about which comes first between the two, it only starts to matter once you start hitting the intersection between the two, which is very theoretical and not much use to designers.
The nice thing about an ME degree is that you touch every single area of engineering, they essentially educate us to the level of a third year electrical engineer by skipping the theory. Because of this, I had the knowledge necessary to prototype LED lighting systems at my internship, and spec equivalent lighting for mass production. I could also use my programming and calculus knowledge to generate algorithmic designs that would have been extremely tedious to do by hand.
Hope that sheds some light