Can 的 be omitted after 谁?

谁 is an Interrogative Pronoun. When 谁 modifies a Noun, we usually followed it with 的.
eg,
谁的书？
比一比谁的成绩好？
这是谁的意见？

And if the Noun is a kinship Noun, the 的 can be taken with or without.
他是谁的哥哥？ (ok)
他是谁哥哥？ (ok)

老师 is a Noun, but it is not a kinship Noun, so 的 should be added.
他是谁的老师？ (ok)
他是谁老师？ (X)

Generally correct grammar includes 的 after 進 because 的 also acts as a possessive grammatical composition. "進的..." translates to "who's..."

well ,this is a good question ,中英文刚好有对应的词，理解起来更方便。
1. Both Chinese and English just have the corresponding words, so it is more convenient to understand.
when you ask question using by whose, you cant us "who ", this is exectly like that in chinese,
whose book is this? 这是谁的书？
who is he? 他是谁？
2, in the family relationship, omission of "的"because the relationship is natural and inherent , different from the affiliation of stuff ,so , in chinese we ommit "的", however , when we say 我的书，it cant ommit "的" since theaffiliation is not so close like family.
3 the another reason to omit because too much commenly used, this linguistic phenomenon exists in many languages, we can broke the rule , in english, for example : we should say: let's to go but we say let's go. omit "to "
in chinese, we should say : 被太阳晒 实际上， 我们说： 晒太阳 passive is omitted.
this is for your reference only

yes，right.If talking about close relationships,you can omit 的。like，中国的北京——中国北京。 我的妈妈——我妈妈。 谁的 means whose ，you don't know their relationships, so it can't be omitted.

结构助词

“的”作为结构助词，往往都是连接修饰限制语和它的中心词。例如：美麗的天空。
我爸爸是因為我已經說明是我的，不需要贅述一次。誰的老師就不一樣了，因為我不知道是誰的。

if its oral Chinese, its okay if u omit 的.