How to properly pronounce コップ (meaning: cup)? Should it be コ + short and quick ツ + soundless プ or something different？ BTW, is there a good online source where I can look up word with proper Japanese pronunciation?
As for the pronunciation, read as it is written in katakana. There is no special rule you have to consider.
However, both コップ and カップ are commonly used in Japanese, and that may be the source of confusion. So let's see the difference.
Google's image search would be the best tool to grasp the idea. I think you have to just memorize when to use コップ and when to use カップ...
(This is intended to be a "simple answer". I assumed the OP does not just want to know how to pronounce コップ, which is relatively trivial - see note - but when to use コップ and when to use カップ, which is confusing.)
There are two similar words with different meanings - コップ and カップ, kOppu and kAppu.
It is typically used to refer to
This is usually a
According to Wikipedia, コップ comes from the Dutch word for cup, "kop". You can consider it one type of カップ.
The same phenomenon applies to the word for glass for which there are two words:
This time ガラス comes from the Dutch word "glas", グラス comes from English word "glass".