Can I copy an image from another paper (that I am citing)? I am in the process of writing a paper on the results I have obtained recently. One of the steps where I have innovated is extracted from a paper published some years ago.

My idea for the paper was to compare the original method with my modifications and thus present the different results.

Is it generally accepted practice to simply copy the "results" image from the original paper (that I am extensively citing and discussing in mine)? Or should I somehow recreate it? (this last option is presenting some difficulties, as some implementation details* are not explicitely stated in the original paper and thus I cannot be 100% sure that my reproduction would be actually representative of what the authors originally did)

*: yes, we are speaking (also) about code
You have two different questions: one in your title, regarding coping an image, and another regarding the results.

Copying a figure:

It depends on the license of the paper. If it is appropriately licensed (as with a Creative Commons Attribution license), you generally can, as long as you indicate it. If it is copyrighted, you are in the grey area of possible fair use. In these cases, the safest option is to contact the copyright holders (usually the publisher) and ask for permission.

Another option is to contact the authors and ask them for the raw data itself, so you can plot it yourself (so you keep a constant style across the paper), or ask them to regenerate it for you.

Using results:

The results are not copyrightable, so you can freely use them. If Smith et al, 2007 report an accuracy of 91%, and you get a 97%, you can freely put the numbers next to each other.