Will floating point code return the same arithmetical results on two different computers?
As long as you execute the same machine code on the different machines and as long as the settings for the floating point unit are identical, you will get identical results. However, you cannot execute the same machine code on both Intel and ARM, so this answer is only hypothetic. Even on different Intel processors you have to take special care that exactly the same machine code gets executed. Some third party libraries like Intel-MKL use different machine specific code on different processors, and other libraries like FFTW measure speed at runtime, and select different algorithms (and hence different code) based on the outcome of these measurements.

Sometimes the compiler is able to inline a certain function (for example your rounding logic), and then you can end up with different results for identical input to that function at different parts of your code. Another very real non-reproducability for distributed floating point computations are potential reductions are the end of the computation. Because the result here depends on the order of the reduction steps, it is very easy to end up with slightly non-repreducible results.

I have run into the issues in the past, but never considered this to be a reason to switch to fixed-point algorithms. For robustness problems of "computational geometry" related code on the other hand, I consider integer or fixed point arithmetic to be an appropriate solution.


  Should I use fixed-point algorithms to circumvent this possibility?


In conclusion, there are several issues related floating point numbers, and even some problems where floating point numbers are not the most appropriate solution, but in general floating point arithmetic is still the way to go. So no, you should not abandon floating arithmetic in general, but only in the specific cases where it causes unsurmountable robustness problems.