As I am a newbie in DSP and I thought this article has a clear explanation, I did not try to answer your questions myself. But you said you have read it and thought these questions are still ambiguous, then I want to try one more time to give my answers.
First, about the concepts of bins and bands, they confused me a lot in the beginning when I was trying to figure out what the Fourier Transform means, especially the Discrete Fourier Transform (DFT), which is also much related to your questions. Right now my understanding of frequency bins are specific frequency values, such as 2000, or 2001.5. A frequency band, for example from 1000 to 2000, covers a range of frequencies, and it is defined by the two frequency bins 1000 and 2000 (one is the floor, and one is the ceiling). In the physical world, the values of frequencies can take any real number, in other words, they are continuous. As to the above example, there are infinite values or frequency bins in the frequency band [1000,2000]. However, when comes to digital signal processing, in the time domain, we take finite (and integer) numbers of samples with a sampling rate, for example 44100 samples per second (i.e. 44100Hz). Later we do Discrete Fourier Transform (DFT) in ways of FFT or STFT to go to the frequency domain. When we do STFT, we do it in the unit of frame and apply hanning window to it. As you mentioned the width of the window equals to the frame size such as 2048 samples, and 2048 is your N.
As to your questions:
It is not that we further partition a frequency scale into bins, naturally we get N/2 frequency bins after FFT or STFT. If N = 2048, sampling_rate = 44100, then we have 1024 frequency bins [21.53, 43.07, 64.60, ..., 22028.47, 22050]. Our finite and discrete sampling in the time domain has already limited the frequencies we get to be finite and discrete too, as bins.
As explained above, usually we filter(maybe can be understood as combine or group) the frequency bins to get bands.
Naturally we have more than one way to group bins to bands, I think the triangular filter is just one specific (Mel) way to get Mel spectrogram.
As to this question, the above tutorial definitely explains better than me.
About the question 3, I am not sure on what level you are asking about the reason to do the triangular filter. If it is about the choice of the specific technique/algorithm, I think it is related to the question 4, and I will leave it to the tutorial. If it is about the bigger reason to do Mel transform, this comes to the field of human sensation and perception. When we talk about frequency and amplitude of the sound source,these are pure physical features of sound. Pitch and loudness are their corresponding human perceptual features. The point is our human perceptions do not linearly change with the values of the physical features. That is the reason we have Decibel scale and Mel scale, therefore our perception change linearly with the values in these scales. We only need to do some math transforms to find the real physical values corresponding to these scales or vice versa.
I hope these can offer some help.