My dictionary does not even have the word
Kärtchen, and online translators have very little about this noun. I think that it's neuter (das), yet I would like to know if this word has a plural form and if so, how it is spelled.
One reason you are having difficulties finding an adequate dictionary entry is because Kärtchen is what is known as a diminutive form. One of the diminutive endings -chen or -lein, when added to nouns, makes the modified noun smaller, cuter, or (less often) laughable. (Unfortunately for many words one of the two diminutives is much more common than the other, but there is no easy rule for it.)
Take for example:
Das Tierchen (the little animal)
Das Brötchen (small bread, here: a roll)
Das Häuschen (the little house)
You'll notice that all the diminutives are assigned the neuter (das) article. Additionally, all diminutive forms remain the same in the plural as their singular. Thus,
Die Tierchen (the little animals)
Die Brötchen (the rolls)
Die Häuschen (the little houses)
In the case of Kärtchen, this is the diminutive form of die Karte. As you can see, some vowels (a, o, and u) become the umlauted diphthongs (ä, ö, ü) when the diminutive is made. Additionally, nouns ending in -e often (but not always) drop this letter. Both of the above morphological phenomena can be seen in the example of die Karte -> das Kärtchen.
Following the above rule regarding the pluralization of diminutives, its plural is die Kärtchen. Since every noun has a possible diminutive form, most are left out of the dictionary. The only ones that are found in the dictionary are typically ones with concrete meanings such as Brötchen, Mädchen (girl), etc.
Not every word that ends in the letters chen is a diminutive variant, as Alexander has pointed out in the comments. A few examples:
Eichen (oak trees)
BUT Eichen (little egg[s]) from Ei + chen
Aachen (a city in Germany)
This form is build from a noun and a suffix like: -lein, -chen, -ette, -erl.
One approximation to get the root word:
- Remove -lein, -chen, -erl, -ette from the end of the word.
- Replace ä with a, ü with u, ö with o and ß with s.
If the word is a "zusammengesetztes Hauptwort" only replace the letters of the "Hauptwort" at the end.
- Add -e, -er, -en or nothing to the end of the word.
This does not cover all possibilities.
An example where this approximation is wrong: