Kinagara?

We can say shinagara but can we say kinagara too? Like I was doing...while I was coming.

No, we need to rephrase it to say "on the way coming (here)," "(kochira e) kuru tochuu." This is because the verb "kuru/kimasu" is a verb that shows a movement, and cannot quite be used in the "-ing" tense as in English. ("suru/shimasu" can be a continuous action)

It is not easy to logically explain the difference, but we use this "nagara" when a verb means or implies a situation, a continuous action (e.g. "miru" [watch]) or a chain of actions (e.g. "kiru" [wear, put on]), not when a verb means a movement but does not imply its procedure (e.g. "kuru").

"Kinagara" usually means "kiru" + "nagara" [while putting on], not "kuru" + "naraga" [while coming].

"Ido(u)suru" can be used in the both ways.
e.g.
(1) Kare wa Tokyo e ido(u)shinagara kazoku no koto wo kangaeteita.
[He was thinking about his family (simultaneously) while he proceeded to Tokyo.]
(2) Kare wa Tokyo e ido(u)suru tochuu de kazoku no tame ni omiyage wo katta.
[He bought souvenir for his family on the way he proceeded to Tokyo.]
"Ido(u)shinagara" does not suit the case (2), because he stopped proceeding to Tokyo for a while in order to buy souvenir, i.e. he did not proceed and buy simultaneously.

You might encounter an example where "nagara" means another.
e.g.
Kanojo wa kotae wo shitteinagara damatte ita.
[Although she knew the answer, she kept silent]
Karera wa machigai ni ki ga tsuiteinagara sore wo naosanaktta.
[Despite they were aware of their mistake, they did not correct it.]

NOT: naosanaktta.
BUT: naosanakatta.

Sorry for another error.
Let me replace "Despite" by "Although."
(I tried to convey "despite the fact that....")

Arigatou gozaimasu, wakarimashita!