> Do songs that are in the key of 'C' always start with a 'C' chord?
No, not always. I'd say 95 % of songs started with the root chord, that is, starts with a C chord if the key is C major and an Am chord if the key is A minor.
A better bet is the last chord. Probably more than 99 % of songs end with the root chord so it's a safe bet that the song is in key of C major if it ends with a C chord. If it does not end with the root chord, you will feel "un-resolved" (left hanging, without conclusion).
I initially posted from work and may not have been as specific as I should have been.
I'm downloading songs/lyrics/tab etc.
My uke group (SOUP...Southern Ontario Ukulele Players) has a playlist of about fifty songs. They play in the key of C. I have been downloading versions of the playlist songs, and upon getting back to the weekly jam session realized that I had downloaded versions of the songs on the playlist in keys other than C.
I will examine the songs that I have acquired according to your criteria.
There are a lot of songs out there in the Key of C but the ones that aren't can be easily transposed into C. I do that a lot because I find C an easier key to play in. If you use the I IV V7 chord progression its easy to plug any thing in within reason. Some songs won't sound right in C and you can tell that by playing them in C. Some musicians like G or D etc but those songs can be transposed to what ever you like.
I have found that the last note in a song is usually in the song's key with some exceptions.
Any song can be in the key of C if you put it there.
For me, a better way of determining key is looking at all the chords in it... not just the first and/or last. (Although first/last really does help.)
The basic chords that naturally occur in the key of C are C, Dm, Em, F, G, Am, and Bdim. (You might also find extensions on them... those tacked-on numbers like C6, G7, Am7, etc.) If all the chords in a song are in that list, then the song is typically in the key of C.
I got more into the whole key/scale/melody/chord relationship in this recent post.