the answer has a quantum nature of uncertainty: for every person in the world at different times of his life and at different stages of his development, he has a range of options from complete inadmissibility to absolute norm. 1) yes, it is POSSIBLE, no problems. I'm sure I'll surprise someone, but you can even wash your hair with dish soap. all detergents intended for the skin are also suitable for the hair. this is a simple approach. it is not suitable for those who are passionate about their hair more than just keeping it normal. 2) IN NO EVENT! a more "sophisticated" case when you can not wash your hair in any case with something that is not directly intended for this purpose and for this there is no indication of at least a psychic, and better - a personal beautician, whose bills not only upset your budget, but also raise a sense of self-importance with each visit.
I will not pay too much attention to the second pathology, but I will return to the first option - this is my answer and it is absolutely sound, tested not only by theory, but also by many years of practice - just water , laundry / hand / baby soap, shampoo, shower gel, dishwashing detergent and even an average solution of various detergents based on different anionic surfactants (with a pH close to normal).
p.s. by the way, pH is an important factor, so you can also wash your hair with washing powder, but very rarely and, preferably, diluting it strongly with water
Gels and shampoos usually have the same sodium lauryl sulfate or something similar in the base. Personally, I have been washing my long hair for several years only with gel, and everything is fine with it. And from the additional components of the shampoo sensations there is something itching, or some other garbage. So there will be nothing wrong with the gel, and the conditioner may simply not clean them well enough.
If you wash your hair with a shower gel, nothing like that will happen (several times the shampoo suddenly ran out, the shower gel saved me a lot)). Hair will only creak unnecessarily. More than after shampoo, because (I guess) there is no silicone in shower gels, which gives a "lubricating" effect like the light version of conditioner. But I recommend using the conditioner after the gel. And for shampooing with balms and conditioners, there is a whole special term - cowoshing. Some ladies like him too. True, it is advised to alternate with washing with normal shampoo.