I know that onces a software company makes any software, the aesthetic look and UI look of that software is automatically copyrighted, and thus any other company (read I) cannot copy it.
Anyhow, what I want to know is as to whether I can copy parts of the UI of an old version of a software made by a rival company (i.e from a very similar software in thers of functionality, made by another company)?
Your first paragraph is at least partially incorrect, in the US. See the US Supreme Court ruling on Borland vs Lotus. It all depends on what you are copying from the look and feel of the program.
You copy at your own risk. Even if there is some legal ground to stand on, as the answer above suggests, competitors sometimes do whatever they deem necessary to protect their products.
If the owner of the material discovers your copying, you are likely to receive a letter from an attorney. There are likely to be grounds to file a complaint and perhaps subject you to several months of interrogatories, document requests, and depositions. This will present a serious disruption and misdirect important resources.
Also, copyright may not be your only concern. UI is increasingly becoming the subject of design patents.
Wholesale copying is rarely a good idea. Even when legal, it looks bad. If not legal, then you are facing enhanced consequences. If you really need to base your product on a competitor's, then consider consulting an attorney to figure out a design-around.