All speakers have a little EQ built in

Why not, as long as you avoid making adjustments that are too high or too low, which could cause clipping, and the quality of the sound such as dynamics and microdetails remains unaffected. Especially during low-volume listening, it is desirable to apply EQ adjustments to compensate for highs and lows to achieve an equal loudness contour.

However, the questions that concern me are:

1. Will the quality of sound genuinely remain unaffected with EQ?

2. Should I opt for digital EQ (especially PEQ) or analog EQ?

Although many people attest that the analog EQ does not alter the sound trait in general, I am suspecious if that is true. Essentially, all analog equalizers were electronic circuits with inductors and capacitors which shift the phase of AC signals passing through them to alter the frequency response in different bandwidths. These physical electronic components have the potential to influence sound quality as @mashif has mentioned. For instance, a larger capacitor tends to perform better than its smaller counterpart, producing a bolder and cleaner sound.

On the other hand, the PEQ allows for the adjustment of the Q factor, enabling tone shaping. In contrast, analog EQ is fixed with a Q factor usually set around 1.5. When used judiciously, the digital EQ tool tends to do a better job of tweaking the sound to my liking compared to analog EQ. I rarely notice any degradation in sound quality with the digital EQ.