I do not use Dropbox, so cannot help with that.
However, the first two items here seem to be expected behavior.
If you click the Finder icon in the Dock (far left) any Finder windows already open will come to the front and "Finder" will appear in the menu bar next to the Apple logo (which is on the far left). However, if there are not any Finder windows already open, nothing else will change, and the only way you can tell that the Finder is the frontmost and active app is that "Finder" is displayed next to the Apple in the menu bar.
If you want to switch to another app (including Finder) and hide all other windows, press Command-Option while clicking on that app's icon in the Dock. This helps me sometimes.
If Safari is the active (and frontmost) app, "Safari" will appear next to the Apple icon. If you have closed all Safari windows (by clicking the red button or pressing Command-W), Safari will continue to be active and if you press Command-N while in Safari you will get a new Safari window, not a new Finder window.
If all other windows are closed, and Safari happens to be the active app, this can occur even though nothing but the desktop is in view.
It sounds like you might be in Safari with no windows open. True?
What happens when you hold Cmd-Option and click on Finder icon, then press Cmd-N?
Hope this helps.