To answer the initial topic, no, your English degree won't make much of a difference. There are plenty of people on JET with degrees in engineering, science, math, business, and politics (and whatever else I'm missing). Sure, an English degree is great to have because you're an expert of the language. You can definitely answer more technical questions that the JTE's (Japanese English teachers) might ask you, which is great. Unfortunately, the technicalities of English are unnecessary. The goal of English in Japan is not to make everyone fluent speakers and writers. Some would say that ALT's are here to give Japanese a chance to speak English with a foreigner and improve their English. Others might say that ALT's are here to share their own culture with Japanese, and to go back home to share Japanese culture with their friends and family. Those reasons are valid, and there are probably dozens of others reasons that are just as valid.
Although, one thing that most people don't know is that the JET programme's mandate is, first and foremost, to promote internationalization. This can be found in the JET handbook and (iirc) on the CLAIR website. This is why an English degree won't make you a better candidate than somebody else. A wise friend once told me that a degree is a piece of paper that tells an employer that you're able to commit to a long-term project and finish the job (ie. the 4 years of school). I like to believe that the JET programme/CLAIR looks at a degree with the same sort of perspective.
Jumping to your other question/topic. There's no placement in Tokyo at this time. Although, there is hope. There are plans to increase the number of JET programme ALT's in Japan, and there are also plans to add positions in Tokyo. That's the good news. The bad news, is that they'll probably place the more qualified ALT's/teachers there. IE: people with teaching degrees, people with a high level of Japanese, people with lots of teaching experience, etc. <--- That's just my theory though. I can be wrong, but that's what I've dissected from the time that I've been living in Japan.
I noticed that you're also willing to be placed in Sapporo; that's definitely an option. I think that your chances of getting it would be pretty good too. Not (as) many people request to be placed in a cold climate with some of the largest snowfall on the planet.
When and if you choose to apply, you can request three locations in which you'd like to be placed. People say don't put Tokyo on the list, but personally, I would. I'm probably already making heads shake, but read on. Us ALT's aren't necessarily more informed about positions/opening. Maybe they'll come to the decision of adding position in Tokyo by May or June. If that ends up being the case, ALT's won't know. Hell, even the JET interview panels won't know. So go on, add it as a request if you want to live there. There's usually a box for you to fill out asking why you want to be placed in location X. If you select Tokyo, just explain that you know there aren't any positions, but that you would like to be placed there if positions open up.
Hope this helps.