Everything (except the things that never shut down because they are important) is shut down. And it will stay that way until Congress can agree on the funding bills (aaand Trump signs it, so there's that). Trump agreed to the two-week continuing resolution, but he can always change his mind. The problem (I believe) is that the House bundled all of the funding bills into a package, so now they have to rewrite it to match what the Senate passed. Then Trump signs it, and it will be the two-week continuing resolution for DHS.
After that, DHS will shut down, unless there is another CR or everyone agrees on a budget for them. And there is enough outrage (hopefully -- us Americans have really short memories) for there to be a radical change to DHS. Bernie Sanders wants to claw back most of the $85B dollars DHS got last year. Other people want DHS to simply cease to exist. The country got along just fine before then. It only came into existence after 9/11, and it really isn't needed for anything.
Whatever the House comes up with, it will have to pass in the House and then pass in the Senate. Passing anything in the House is (mostly) easy because they only need a majority, but the repubs in the Senate don't have the 60 votes to get it through cloture, so it needs to be something that the Senate dems are willing to go along with.
If this wasn't the case, the repubs in the Senate would ram through the DHS budget that the House already passed. Thus, the "Democratic led shutdown" of DHS. Assuming Johnson can get the House to pass the Senate's version quickly.