1
Use the menu > Objects > Objects...
then a new window will open with a pretty list of all (drumroll) objects.
When you select in the list, the actual object gets marked in your illustration or document.
When you are interested in "managing" your objects (maybe you got complex documents) you can right-click any object, go properties and assign your own labels; which you will then find in the object list.
About the quantity of objects, @Juancho has already hinted at selecting and finding the total in the status bar. If your document is very complex, you might prefer using the menu again > Edit > Select All so that you will not miss any tiny or colour-less or off-page objects.
2
Select the object you want to treat in isolation. Then use the menu > Edit > Invert Selection
Now open your object list and you will find a blue block of selected objects. Click on any of the eyes on the leftmost column and they will all be invisible.
This is not a separate window, sorry, but my best answer to your request. Since objects can and do interact (for example overlap) in Inkscape I do not see how you could have a single object in a separate window and do meaningful work on it. Maybe I am wrong, but it feels like a surgeon saying "I would rather take this brain to my lab, than work with all this bloody mess in the operating room with the rest of the body attached."
And no panic, about making all your stuff visible at the end: Select All from the menu will only work on visible objects, i.e. will not help you here. You should rather go into the object list and use CTRL+A (to select all in that list) and then use the eye at the very top (typically for an entire layer) and click that to make all objects visible.
