A bounding box is a rectangle superimposed over an image within which all important features of a particular object is expected to reside. It's purpose is to reduce the range of search for those object features and thereby conserve computing resources: Allocation of memory, processors, cores, processing time, some other resource, or a combination of them. For instance, when a convolution kernel is used, the bounding box can significantly limit the range of the travel for the kernel in relation to the input frame.
When an object is in the forefront of a scene and a surface of that object is faced front with respect to the camera, edge detection leads directly to that surface's outlines, which lead to object extent in the optical focal plane. When edges of object surfaces are partly obscured, the potential visual recognition value of modelling the object, depth of field, stereoscopy, or extrapolation of spin and trajectory increases to make up for the obscurity.
A classifier can only classify one object per image
A collection of objects is an object, and the objects in the collection or statistics about them can be characterized mathematically as attributes of the collection object. A classifier dealing with such a case can produce a multidimensional classification of that collection object, the dimensions of which can correspond to the objects in the collection. Because of that case, the statement is false.
1) If I don't need to know 'where' an object is (or objects are) and just interested in the existence of them and how many there are, is it possible to just get rid of the boundary boxes?
If you have sufficient resources or patience to process portions of the frame that don't contain the objects, yes.
Questions (2) and (3) are already addressed above, but let's look at them in that context.
2.a) If not, how does bounding boxes help detect objects?
It helps by fulfilling its purpose, to reduce the range of the search. If by thrifty method a bounding shape of any type can be created, then the narrowing of focus can be used to reduce the computing burden on the less thrifty method by eliminating pixels that are not necessary to peruse with more resource-consuming-per-pixel methods. These less thrifty methods may be necessary to recognize surfaces, motion, and obscured edges and reflections so that the detection of object trajectory can be obtained with reliability.
That these thrifty mechanisms to find the region of focus and these less thrifty mechanisms to use that information and then determine activity at higher levels of abstraction are artificial networks of this type or that or use algorithms of this type or that is not relevant yet. First understand the need to reduce computing cost in AI, which is a pervasive concept for anything more complex than tic-tac-toe, and then consider how bounding boxes help the AI engineer and the stakeholders of the engineering project to procure technology that is viable in the market.
2.b) From what I have figured is that a network (if using neural network architectures) predicts the coordinates of the bounding boxes if there is something in the feature map. Doesn't this mean that the detector already knows where the object is (at least briefly)?
2.c) So continuing from question 1, if I'm not interested in the exact location, would training for bounding boxes be irrelevant?
Cognition is something AI seeks to simulate, and many hope to have robots like in the movies that can help out and be invaluable friends, like TARS in the Nolan brothers 2014 film Interstellar. We're not there. The network knows nothing. It can train a complex connection between an input signal through a series of attenuation matrices and activation functions to produce an output signal statistically consistent with its loss function, value function, or some other criteria.
The inner layers of an artificial net may, if not directed to do so, produce something equivalent to a bounding region only if velocity of convergence is present as a factor in its loss or value function. Otherwise there is nothing in the Jacobian leading convergence to reduce its own time to completion. Therefore, the process may complete, but not as well as if cognition steps in and decides that the bounding region will be found first and then used to reduce the total burden of mechanical (arithmetic) operations to find the desired output signal as a function of input signal.
3) Finally, in architectures like YOLO, it seems that they predict the probability of each class on each grid (e.g. 7 x 7 for YOLO v1). What would be the purpose of bounding boxes in this architecture other than that it shows exactly where the object is? Obviously, the class has already been predicted so I'm guessing that it doesn't help classify better.
Reading the section in A Real-Time Chinese Traffic Sign Detection Algorithm Based on Modified YOLOv2, J Zhang, M Huang, X Jin, X Li, 2017, may help further comprehension of these principles and their almost universal role in AI, especially the text around their statement, "The Network Architecture of YOLO v2 YOLO employs a single neural network to predict bounding boxes and class probabilities directly from full images in one inference. It divides the input image into S × S grids." This way you can see the use of these principles in the achievement of specific research goals.
Other such applications can be found by simply reading the article full text available on the right side of an academic search for yolo algorithm and using ctrl-f to find the word bound.