No, you will definitely still need to apply for a work permit to teach English in Japan. The new resident card system will come into effect from 9 July this year. Instead of having to report at a local municipal office when you arrive in Japan, you will be issued the card at certain airports (Narita, Haneda, Chubu and Kansai). The new card looks a bit like a driver's licence and will have a photo of you, your residential address, date of birth, resident status and period of stay (in other words, how long your working visa is valid for).
Getting a visa is a tricky business, made much quicker and easier if you secure a job offer from a company in Japan before you apply. The company will help you organise a Certificate of Eligibility, which you can submit with whatever other documentation you need for your particular visa. The specifics will depend on what visa you're applying for (i.e. what work you intend to do in Japan). An English teacher will be a "Specialist in Humanities" and to apply you'll need degree certificates, a police clearance certificate, reference letters and a career history, amoung other documentation. Check out our Visas for Japan page for more information.
Not at all. You can return to Japan in December - as long as you return to Japan before your Alien Registration Certificate expires, and as long as you already hold a multiple-entry visa.
The new card is now issued at airports, so it's possible that you will be able to organise one on your way back into the country. You could try contacting your local Japanese embassy to ask whether this is the case.