What do で and ように express here?

Question

For full context: http://www3.nhk.or.jp/news/easy/k10011221251000/k10011221251000.html

The sentence in question:

ＪＲ東日本は、新幹線で無線のＬＡＮのサービスを始めて、無料でインターネットが利用できるようにすることにしました

My problems are the parts in bold. First, my attempt at translation:

JR-East-Japan has decided that, establishing wireless LAN service in shinkansen, they make it so that you can use the internet for free.

So, first there 新幹線で and 無料で. In the way I translated these parts, it feels like I'm wrong. Whenever で is a particle that marks a location, it usually was in context of an action happening in this place. In this case, "establishing WLAN" can be interpreted as an action happening there, however, the cases I used to encounter with で expressing the semantics I mentioned above were more of this sort: キャシーがパーティーでなに何をき着ていたかおぼ覚えていますか。 And I think that this is something else, although I could be wrong as well of course.

Second, 無料で. It's a similar problem here. I interpreted 無料で as 無料 + conjunctive form of だ. In this case, the literal translation would be:

...being for free, they can use the internet.

However, this already expresses something else than what I translated above, I think. I can't think of another way to translate this meaningfully, and it fits into the context very well.

Third, ように in インターネットが利用できるようにすることにしました. I translated it in a final sense like in "They do X, so that they achieve Y." However, combined with することにしました it sounds rather overcomplicated to me: "They decided that they do, so that X/in order to X...". That's why I feel the need to ask for confirmation...^^ I also know a different meaning for verb+ように, which would be something along these lines: 学生に図書館で物を食べないように注意しました。 Here, my textbook says that "this structure is used to quote the contents of an order or request.". Since sometimes the definitions turned out be defined too narrow, I wanted to bring this up in context of this question.

Answer 1

新幹線で

This is just representing the location of 始まる and explains where the action will occur. Your confusion may have been caused by the often ill-explained difference between に and で when marking location. JSL schools tend to teach that で marks the location of action and に marks the location of existence but it seems that a better interpretation is that に marks the location when it is critical to the verb and で when it is not. See に and で revisited.

無料で

One of the uses of で is to indicate cost. I'm not certain and I hope someone else will address the question of whether a native Japanese would consider this で a case particle or the て形 of だ／です which can also be interpreted as an adverb and in this case describe the extent of the verb.

利用できるようにすることにしました

I'm going to try explaining this in pieces without specifying any nouns.

Aにしました
Someone decided A (as you've identified)

Bすること
Doing B

利用できるように
So that someone is able to use something (indicating purpose as you guessed)

So all in all, someone decided to do something so that someone is able to use something

ＪＲ東日本は、新幹線で無線のＬＡＮのサービスを始めて、無料でインターネットが利用できるようにすることにしました

JR East-Japan is beginning wireless LAN service on the shinkansen and decided to make it so that passengers are able to use the internet for free. (This could obviously be worded better in English but I'm trying to make a point).

As usual, I welcome comments about any errors I may have made. I am merely a student after all.