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.
- コップ: pronounced as KOPPU. This roughly corresponds to a mug, but can refer to a cylinder-shaped cup made of glass like this. A paper cup is called 紙コップ.
- カップ: pronounced as KAPPU. Used for コーヒーカップ、ティーカップ、マグカップ、etc. Interestingly, a mug is コップ, but マグカップ is more commonly used than マグコップ. I don't know why. And World Cup is ワールドカップ, not ワールドコップ.
- グラス: a drinking glass. ワイングラス, カクテルグラス, etc. Glass as material is called ガラス.
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.
カップ (kAppu) is the general word,
It is typically used to refer to
- cups that go with saucers, such as a tea-cup,
- cups you win, such as the Soccer World Cup
- most other cup related derivatives such as brassiere cup (ブラジャーのカップ), cupcake (カップケーキ) or pot-noodle (カップヌードル)
コップ （kOppu） refers to cups for drinking
This is usually a
- glass tumbler ((ガラス)コップ) but also includes
- paper cups (紙コップ） and plastic cups (プラスチックコップ）
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:
ガラス (gArasu) ＝ the material itself (hence ガラスコップ)
グラス (gUrasu) ＝ the drinking cup made of glass, including ガラスコップ and ワイングラス
This time ガラス comes from the Dutch word "glas", グラス comes from English word "glass".