Vilken does mean which, but it is also often used when what would be used in English. We never use vad in expressions with a noun, so it must be vilken tid.
If vad is used with a noun, we must add the preposition för between them, but then it will mean more like 'what kind of'. Vad för bil har du? 'What car do you have?'
No "finns" means to exists, and you will probably exists even if you are busy :-)
Either "När är du tillgänglig" = "When are you available" (a bit formal), or "När kan du" = "When are you able". (In English it sounds a bit odd without specifying e.g "to meet", but in Swedish it sounds okay as written (if we know what is meant)). "Vid vilken tid" may replace "När" in both.
However, one can use "finns" in for example "När finns du på plats?" - "When are you present?" (literally "på plats" = "at place")
The alternate translation was given as "By what time", but those 2 questions mean different things in English. "By" implies a deadline, whereas "At" implies an action will occur at a specific time. I am not certain which is the correct translation, but they should not both work, unless a single Swedish phrase is used for 2 different concepts. Can you please clarify for me?