While I don’t have enough working experience with GIT (I’ve been using SVN in most of my recent workplaces) the process you describe sounds more fitting to a centralized source control system (such as SVN). A distributed source control like GIT means a different way to do things.
In any case both tools are used widely enough to be supported by almost all CI servers out there.
Regarding your CI,
I think you should check out TeamCity before you go down the Jenkins road. Jenkins is more fitted to the Java world and less to the .Net.
Also, I urge you to try avoiding MSBuild (or any XML based build technology). A visual based build technology (FinalBuilder Visual build,…) is much more cost effective in the medium-long run. While at the start (where you build is relatively simple) the difference will not seem to justify the costs involved down the road (and not too far) the ROI will be positive.
And last, I think that setting up a CI server and starting your automation road is actually more crucial than the exact source control tool you will choose (given that you will use one).
Hope this helps