Finding crosses and their orientation on a given image
Try to use cv2.HoughLinesP (see an example) to detect the straight lines (segments). As a result you will get the segments' endpoints, so finding all the intersections and its type will be quite simple maths.
The problem will be rather with finding the proper and universal parameters.

Theoretically you could use a morphological operation called hit-and-miss transform, but this will be difficult in your case, as the lines are hand-drawn and even after thinning the crosspoints may be difficult to be properly classified (as 'T', '+' or 'corner').

I would suggest you to make some morphological preprocessing to improve the binary image you have got.