There is a saying that we Shia have in our Iqamah and Adhan like "Ash hadu anna Aliyyan waliAllah" and similar sayings. What is the ruling on saying these and where did it come from? Also, what is the proper thing to say?

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Sunnis added in Adhan Salaat khayrun minaanoum
Shias added the third testimony.
That is an indication that Muslims can add an important message is adhan as long as they do not consider it an obligatory part of adhan. And our maraje do not consider it as a part of adhan.