"I have to report to my boss on my work every month."
Yes, you are right. However, depending on the context/conversation, the sentence becomes less specific.
If the sentence excludes 都, then "every month" is being specified generally. It does not exactly indicate that the subject is doing the job all these times. Otherwise, the subject does the job "every single month".
Not necessarily here, but in many cases it will be, as 每...都/也... works as a set pairing with both halves required, similar to "neither...nor..." in English.
If I use the word order 我要每个月... does that change the meaning, make it ungrammatical, or is it also acceptable?
Hm... Does this change slightly shift the focus? I mean, the adverb applies to the verb it follows. So in case of ...要每个月... we just describe the routine, "I need to submit monthly reports", whereas with ...每个月要... we rather demonstrate our attitude to the event, like "every month I have to submit the [damned] reports!" Or am I seeing too much here?
Without context it is impossible to tell where the real emphasis is for these two normal sentences. For example, both of them can be naturally followed by 还要… (“also have to …”).
- 我每个月要向老板汇报工作，还要(每个月).跟.客户.联系。 (maybe both tasks are monthly)
- 我要每个月向老板汇报工作，还要(每个月).跟.客户.联系。 (maybe both tasks are monthly)
- 我每个月要向老板汇报工作，还要(每个月)给他.清理.一次.电脑。 (more likely a complaint than the two sentences above)
- 我要每个月向老板汇报工作，还要(每个月)给他.清理.一次.电脑。 (ditto)
Not sure, but it sounds rather casual to me. It is like omitting the preceding sentence, e.g. “不管怎样… (Anyway …)”.
It also sounds like “Even I have to …” if the context is not clear enough.