Eagle will generate Gerber files which are standard, but it is always a good idea to ask the PCB manufacturer what do they need.
There are some PCB companies that offer PCB application for free, but you need to stick with them.
My recommendation is to use Gerber files since the beginning and double check your design before the final release, nobody wants to spend money because a little mistake on the design.
Yes you can create Gerber files with eagle, You can aslo have a look here
**broken link removed**
They use there own software which is free and then they cna make your boards,
My advice would be use eagle or any othe PCB software you like, First you will have to practice and get to understand how the software works, You will need to get to understand how the schematic and PCb editor works, once you have drawn your schematic check it and double chekc it, becasue if your schematic is wrong then your PCB will be to, Once you know your schematic is 100% then transfer to PCB editor then layout your PCB run the DRC check if this is ok then I would make one at home build it test it, Then once you proved your design is working 100% then if you need lots of them you can create the geber file to send to a PCB manifacture, This way you should not get any extra cost if you found out that your design did not work
Eagle is free if you don't want more than 100mmx100mm board if I remember, There is Labcenter (proteus) which starts of reasonalbe it all really depends how much you want to spend
2. What do you mean by " make one at home build it test it" How will I make PCB at home ??