Why are the lists of Kangxi radicals different everywhere?
There are exactly (no exceptions) 214 Kangxi radicals:





Any misunderstanding to the number of these probably comes from a confusion to what radicals actually are. Remember that radicals are dictionary headers (部首), used to group characters or words in a dictionary. The Kangxi radicals are similarly used to group entries in the Kangxi dictionary, not any other dictionary (unless the maker of another dictionary consciously chooses to use the exact same radical set as the Kangxi radicals). This means that


If a smaller dictionary chooses not to list some of the more obscure characters that are found in larger/more comprehensive dictionaries, then the dictionary may well not have the full set of Kangxi radicals.
The Kangxi dictionary is a Traditional Chinese dictionary, and Simplified Chinese dictionaries will not use the same radicals as the Kangxi radicals (although their entries may be based off the Kangxi dictionary).
Radicals (Kangxi or not) are completely arbitrary. For many characters, they have very little to do with the meaning or sound of the character. It is completely valid for a new dictionary publisher to make up their own radical system to group characters.


To answer your queries directly:


  Whether there are 17-stroke characters or not 


There are incredibly obscure characters which exist in larger dictionaries but are not in common use. Smaller dictionaries for common use may not include these entries - but in any case this has nothing to do with the Kangxi radicals.


  one of the lists regard "老" as a radical on its own, and another treats it as a combination of two different radicals


Two notes here:


Each character has only one radical, just like each English word has only one first letter for which it is grouped under a dictionary
If a dictionary lists「老」under another radical other than「老」itself, they are definitely not using the Kangxi radical system. Here's the page from the Kangxi dictionary itself listing「老」:





  Also, I'm sorry if this question is a bit silly, but does the Kanji in Japanese have anything to do with the Kangxi in Chinese?


No, it doesn't. Kangxi refers to the Kangxi Emperor, the one who mandated the compilation of the Kangxi dictionary. In Japanese, Kangxi is called kōseki, while kanji is the Japanese name for Hanzi/Chinese characters.