I've had a correct solution (courtesy of my friend Chris) and one near miss from a student Joe, so I thought I would give the solutions. First I'll go through the 16 puzzles, then I'll deal with the meta puzzle:
1. Fluffy 
1. FILED Anagram of I FLED.
4. CYMRU My cat is MU + CRY anagram.
5. LUNGS Anagram of SLUNG
1. FOCAL Pure cryptic
2. LEMON Anagram of MELON
3. DRUMS DR + UMS
The yellow squares spell Cloud.
2. Travelling 
Cork - Madrid - London - Barcelona - Brussels
Kassel - Cologne - Strasbourg - Munich - Strasbourg - Nice - Florence
Prague - Split - Sarajevo
Zvolen - Valjevo - Bucharest
Kiev - Rivne - Iasi - Odessa - Ankara - Izmir
Try plotting them on a map:
3. Joanne 
My favourite puzzle. Putting these Harry Potter titles in release order and listing out the languages we get Romanian, Occitan, Welsh, Latvian, Icelandic, Norweigan and German. These spell out Rowling.
4. Left Out? 
Toy, the, wall, cars, ?
These are the first word in every third Pixar film. Toy Story, The Incredibles, WALL:E, Cars II and the 15th was Inside Out, so our answer is Inside.
5. The True Author 
Francis Bacon is one of the favourites of conspiracy nuts to be the true either of Shakespeare's work, but he was also known for the Bacon Cipher where each letter of the alphabet is assigned a five digit set of 0s and 1s. By turning the bold letters into 1s and the unbold as 0s we get a string which translates into Shakespeare.
6. Fantastic! 
dECk Clef - f Stone - e = ECCLESTON (who had the catchphrase "Fantastic" as The Doctor)
7. The First 
You can find more on RegEx Crosswords here. Th top left has to be a letter which is in WHO and that forces the top line to be HA. The rest follows from here giving HARTNELL.
8. Famous First Letters 
Apologies for only giving you 7 out of the 10 letters. Almost every book in my library only provided i or t. You should still have enough information.
The first letters of each novel spell out WO??SWO?TH. I checked so many books making this one. Wordsworth.
9. Feeling Cold? 
I was going for a Goosebumps feel on this one, hence the title, the monsters and the aesthetics. Each line has two words, the first of which has an extra letter. Spider - Pride = S, Troll - Roll (think DnD) = T, Imp - MP = I, Demon - Mode = N, Harpies - Parish = E. Stine was the author of the Goosebumps series.
10. Answering Machine 
I took this one in one take with a ringing sound and the beep coming from well timed YouTube clips that I unpaused and paused at the right points. The NATO PHONEtic Alphabet words can be heard in the clip. Papa, Hotel, Oscar, November, Echo = PHONE
11. Number 
12. Toothy 
It has been pointed out to me that there was an extra character that wasn't intended, but accidently made his way in. Ignore Ted.
I'm going to have to apologise for this one: I misspelled Gunn as Gun. By the time you add in the accidental appearances of characters who were in single episodes this ended up not being the best closed form puzzle I've ever made. The remaining letters spell out Sharp Tooth or Canine if you will.
13. A Man of Words 
Who penned these?
A Man ? and Old
An Irish ? Foresees His Death
Meditations in ? Of Civil War
Leda and the ?
Young, Easter, Airman, Time, Swan. These spell out, and were written by Yeats.
14. Who? 
Toclafane, Empty Child, Novice Hame, Navarinos, Autons, Nyder, Thal. Together they spell out Tennant.
15. Abysmal Elian Script 
16. A Magical Connection 
These are Magic the Gathering Cards and apparently my efforts to distort them were not enough to defeat reverse image search. We have Mage-Ring Network, Aligned Hedron Network, Thopter Spy Network and Psychic Network. So, Network.
Putting them altogether there weren't quite enough Doctor Who actors to make a group. Instead we have:
First Name William: Shakespeare, Yeats, Wordsworth and Hartnell
Known as initials: R L Stine, J K Rowling, H G Wells and H P Lovecraft
Number 9: On Cloud 9, Inside No. 9, Eccleston played the 9th Doctor and IX = 9
Numbers in Words: caNINE, neTWOrk, phONE and TENnant