You have listed only vague ideas in your question, so the answers are somewhat high level.
1) I would closely evaluate if you really need a repository pattern on top of an ORM. What are the chances of switching out EF for another ORM? If you think that is possible, or even possible for your project to completely switch database architectures (SQL to NoSQL?), then go for a repository. Otherwise maybe not.
A repo can add a lot of work for little benefit. You may well find that once you have a well defined code-first model then a repo layer on top of DbSet is turns out to be not much more than a proxy to a very similar call or series of calls in EF. This is because DbSet in EF is already implementing a repository pattern and that may well be enough for your project.
If your model is poorly defined then as your model changes you may find yourself with a lot of work having to change not only the DbSet model and classes that use it but also updating all your repository code and the classes that use that.
Likewise DbContext is already implementing the Unit Of Work pattern for you. Literally the DbContext is a unit of work and its lifetime management should reflect that.
If your main reason for a repository is to abstract the database then this is what EF is already doing for you.
There is always a lot of debate on the value of repository over an ORM like EF. You can find a lot on line with a quick Google and even more on Stack Overflow.
2) While I am totally addicted to DI, I try to use it only where needed because again it can add unneeded extra complexity. The web service invocation seems to be an obvious dependency that could benefit. Injecting a repository everywhere I have my doubts about just because I always have doubts about the value of repository on top of EF. Bear in mind that DbContext should be injected with care and is not thread safe. This article goes into depth on that.
3) TopShelf is my goto when developing Windows services. It makes the development and rollout of them much easier. There is a Nuget package for it. I have never regretted using it.