Questions about progress + where to go from here

I wasn't really sure whether to post in "For Beginners" - according to my profile, at 1420 rating I'm among the top 15% of players on this site, but I really feel as though I'm just beginning to play properly, rather than just making random moves and hanging pieces.

1) After playing very casually since childhood, I started studying chess and improved from ~1200 to ~1400 rating in approximately 6 months. How does this compare to the average? Browsing the forums I've found some people who think that anything below 1600 after 6 months is indicative of mental deficiency, and I've found others who say that most adults can only achieve 1000.

2) What should I do to improve from 1400? Is it time to start expanding my repertoire? As white, I only play 1. e4 - almost exlusively the Italian Game, with the occasional Exchange Ruy. Against 1...c5 I'm somwhat comfortable in an Open Sicilian. Against the French I almost always play the Advance variation, although I've been experimenting with the Exchange French in casual games. As black, I only play 1...e5. I know the Italian Game well, and I know the first 5 moves of other common lines (Ruy Lopez - Morphy Variation, Scotch Game, Wayward Queen Attack). I don't think I have ever won a game against the King's Gambit.

Any help would be appreciated, I just want to become the best player I can be. Thanks in advance

You forgot to say what time control you were talking about. You are 1420 in rapid but only 1185 in blitz, which you have played most often. (1600 out of 2200 games.)

You always have to make clear what group or site you are talking about and what time control. Ratings are not absolute measures like the temperatures that water freezes and boils. They are relative measures- relative only to the particular group where they were estimated.

1420 in rapid isn't bad.

Whoever said that being below1600 after 6 months being evidence of mental deficiency made it up. 

That said, nobody has any real information as to how long it should take to improve. 

Right, rapid is for my serious games. I play an awful lot of 10 minute games because that's where I try new openings / gimmicky trap lines etc. That's part of the reason the rating is so much lower, but also I struggle to play well with such low time controls. I need either 30 min or 15 min with increment to give myself enough time to think about my moves, otherwise I regularly drop pawns and even full pieces.

It is perhaps only a matter of time before hoards descend on you with instructions to ignore the possibility of learning more about specific openings. I am not in that camp, but I think that nearly everyone agrees that openings should not crowd out learning about other aspects of chess play. There also seems to be wide agreement that it is valuable to play over illustrative games. Therefore, if you do want to do more opening learning, I suggest that you do it by looking at instruction with an emphasis on such games. Here are some books with that sort of approach:
First Steps: 1 e4 e5
https://www.newinchess.com/media/wysiwyg/product_pdf/7790.pdf

https://chesscafe.com/book-reviews/first-steps-1-e4-e5-by-john-emms/

First Steps: Queen's Gambit

https://www.newinchess.com/media/wysiwyg/product_pdf/7652.pdf

My First Chess Opening Repertoire for White
https://www.newinchess.com/media/wysiwyg/product_pdf/9033.pdf
https://chessbookreviews.wordpress.com/tag/vincent-moret/
Opening Repertoire 1 e4
https://www.newinchess.com/media/wysiwyg/product_pdf/7819.pdf

In a little while, I will try to come up with some ideas for ways to learn about other aspects of chess.

Tactics is one area where nearly everyone needs more work. You might want to consider books like:

Winning Chess by Irving Chernev and Fred Reinfeld (1948)
https://web.archive.org/web/20140708093415/http://www.chesscafe.com/text/review919.pdf
Back to Basics: Tactics by Dan Heisman (2007)
https://web.archive.org/web/20140708233537/http://www.chesscafe.com/text/review585.pdf
https://www.chess.com/article/view/book-review-back-to-basics-tactics
https://static1.squarespace.com/static/5856bd64ff7c50433c3803db/t/5895fc0ca5790af7895297e4/1486224396755/btbtactics2excerpt.pdf

The value of endgame knowledge is generally recognized.

Silman’s Complete Endgame Course

http://theweekinchess.com/john-watson-reviews/theres-an-end-to-it-all

https://web.archive.org/web/20140708103149/http://www.chesscafe.com/text/review594.pdf

https://www.silmanjamespress.com/shop/chess/silmans-complete-endgame-course/

Possibilities for middlegame help:

The Most Instructive Games of Chess Ever Played by Irving Chernev (1965)
https://chessbookreviews.wordpress.com/tag/most-instructive-games-of-chess-ever-played/
Seirawan stuff:
http://seagaard.dk/review/eng/bo_beginner/ev_winning_chess.asp?KATID=BO&ID=BO-Beginner
http://www.nystar.com/tamarkin/review1.htm
50 Essential Chess Lessons by Steve Giddins
https://web.archive.org/web/20140708100833/http://www.chesscafe.com/text/review534.pdf
http://www.gambitbooks.com/pdfs/50_Essential_Chess_Lessons.pdf

Amateur's Mind by Jeremy Silman
https://web.archive.org/web/20140708094419/http://www.chesscafe.com/text/ammind.pdf
https://www.silmanjamespress.com/shop/chess/amateurs-mind-the-2nd-edition/

Simple Chess by Stean

https://web.archive.org/web/20140708104258/http://www.chesscafe.com/text/review400.pdf

http://store.doverpublications.com/0486424200.html

From your profile you play exclusively rapid and blitz games.  Try to play mostly longer time controls, including "daily" chess, so you have time to think about what you should be doing - speed chess may be fun, but at this stage of your development it will do little to promote your rapid improvement or your understanding of how to play correctly.

It makes sense that taking time to think about what you should be doing would promote improvement in your chess skills.

This is not to suggest that you should necessarily play exclusively slow time controls or daily games, but they should be the greater percentage of your games, much more so than speed games which do almost nothing to promote an understanding of how to play the game well.  Speed chess tends to be primarily an exercise in moving pieces around faster than your opponent while avoiding checkmate, in hopes that his/her clock runs out sooner than yours.

Here's what IM Jeremy Silman has to say on the topic...
https://www.chess.com/article/view/longer-time-controls-are-more-instructive

And the experience of a FIDE Master...
https://www.chess.com/forum/view/general/how-blitz-and-bullet-rotted-my-brain-don-t-let-it-rot-yours

Also, playing without a good foundation in fundamental chess principles and understanding how to apply them will do little to help you become a good chess player. In order to understand these principles it takes study - there is no easy, quick solution to playing good chess....you have to put in the effort to learn...

Good Chess Books for Beginners and Beyond....
https://www.chess.com/blog/RussBell/good-chess-books-for-beginners-and-beyond

Both of those threads are very interesting, thanks!

"... Review each of your games, identifying opening (and other) mistakes with the goal of not repeatedly making the same mistake. ... It is especially critical not to continually fall into opening traps – or even lines that result in difficult positions ..." - NM Dan Heisman (2007)
https://web.archive.org/web/20140627062646/http://www.chesscafe.com/text/heisman81.pdf

Read the reviews before you buy any book.  I know one of those books Russell posts all the time has bad reviews.  He may like that book as a chess instructor, and it may be a good book.  However, as I said, read the reviews and learn for yourself.  

The fact of the matter is no one will be able to tell you how to improve from here.  Most adults when they play chess at a later age do not get much better, ever.  I'm not totally sure about the theory behind this, as, I never see any scientific evidence ever sited when people say most adults can't get really good at chess.  It's all assumptions.  It's possible to get better as an adult, but one thing we know for sure is that it's a lot harder to do as an adult.  Most adults reach a plateau and never leave that rating, ever, and gradually start to get slower as time progresses.  There is a WFM named Alexandra Botez who no matter how much she studies, she is not getting any better, and actually losing rating points.  She is not very old at all.  She is still young, but she thinks it's because of her age that is making her plateau and also starting to get worse.  

I'm not saying it's possible to become World Champion at any age, nor am I saying it's not, but what I am saying is that with age "might" come limits to how far up the rating ladder you can go.  I'm a firm believer in what C.S said, and that is, "There is no goal that is unattainable at any age."  Not an exact quote.  

If you have the right plan you can see yourself improve.  The problem is getting that plan.  I don't think the forums are the best place for that plan.  You can ask, and get some guidance, however, I think it would be best to rely mostly on what Masters or other people you know that are in your life who are strong chess players.  Find out where your local chess club is, and join them.  Ask loads of questions.  Don't ask questions in the forums.  Most people will tell you that you are too old to even think about improving.  

There is some good advice in the forums, but try not to get opinions on how far can you go, and what you need to do to improve.  Most often you will get a book list that may or may not be good for you.