Eagle is rubbish (and I've used it extensively). Its popularity comes more from Linux zealotry than anything else.
If you're using Windows, try DipTrace. It actually works like a modern Windows application, not a 30 year old Motif GUI. ;)
Use Eagle for viewing the Arduino schematics and PCB layouts and then use absolutely anything else to design your own. I would suggest DesignSpark PCB (free) or DipTrace (starting at $75 for commercial or free for non-commercial).
Eagle is absolutely horrible, awful software. I have no idea why people keep recommending it.
I haven't used Diptrace myself, but the feature to move changes to/from schematics does exist:
http://www.diptrace.com/forum/viewtopic.php?f=3&t=9396
You will save yourself a lot of troubles with the layout if you use SMD and not through hole caps for the decoupling capacitors.
The routing mess you are having is because you aren't using a copper pour. Once you do, the caps will have one side sitting o n the copper pour (which will be ground) and you don't need to route tracks around them like that. Seriously, if you are planning to etch that at home, do use a copper pour - if for nothing else then to save you etching time and etchant!
See here how to do it: https://www.youtube.com/watch?v=5bMT8EETyrk
Are you using Eagle by choice or is it company regulated? I tried to get into Eagle but I hated the UI too much.
We used Sprint-Layout at school, which is very basic but also very easy to use. Diptrace seems to be nice as well.
Sort of, yes.
They will accept industry standard files called Gerbers and then do all of the above.
A Gerber file represents a single feature layer of a printed circuit board.
So in a CAD program, you would first layout a schematic, then that would generate a loaf of component objects which you can then physically layout a board.
From that you would generate one Gerber file that represents where the holes are (this would drive a drilling machine), one file that represents the top copper traces, one for the bottom copper traces, etc etc...
It's a lot simpler than people may think.
Have a look at Cadsoft Eagle. It's a very popular program for hobbyists, free up to a certain board size and there's a lot of resources online.
Other options include KiCad (Which is also free and developed by the guys at CERN to make circuit boards for their scientific prototypes) and Diptrace.
There's also a subreddit over at /r/PrintedCircuitBoard.
And I'm happy to answer questions also!
While sometimes it may be easier to make your own circuit board using a DIY method, sometimes this can take a bit of practice and is more difficult the more complex the board gets. Also the design here is double-sided, which requires fairly precise alignment of the two side. This can be tricky and requires a fair amount of practice to get right.
It might be worth looking at local Makerspaces if you go down this route, there'll be a lot of experienced and helpful members among this community.
Get DipTrace. Not only can you change the attributes, value, type and so on of parts while they're on the schematic (or PCB), you can also actually export a BOM without running some horrid ULP.
You can also drop a diode symbol on to the schematic, and then change the pattern (footprint) associated with just that part. For example you can drop a diode symbol on, go pick a SOT23 package, and map the pins to it. It's great for quickly dropping one off parts on the board, or if you've decided to change from one package to another and don't want to tear everything up, re-drop the new package from the library, and then reconnect everything.
Design it yourself! There is not a lot of magic involved in doing so.
Step 1. Find some design software... (Free is good)
http://fritzing.org/ http://www.diptrace.com/
Step 2. Learn how to use it... Its not that hard. Where people trip up is by trying to make the PCB design first... Dont! Start by designing the schematic, then move to the design.
Step 3. Read the datasheets for the IC's you intend to use. The manufacturers give out excellent information in the main. Also look out for the manufacturers "Design Guides". These normally give out PCB design info with example designs!
Step 4. Design your board... Its mostly pick and place...
Step 5. Export the design to Gerber
Step 6. Find a PCB maker and order your boards...
The right PCB maker is a difficult call. If you know how many boards you want that helps as a lot have online quotation systems.
Generally, the Chinese companies will have longer lead times, but are significantly cheaper than local companies. I have used http://www.futurlec.com/PCBService.shtml before without drama.