Some questions about an invstigation

Is "conduct" the correct verb to use whit investigation? For example can I use also carring out or perform?

Conduct is both a 'verb' - to conduct/carry out and a 'noun' - he was report to the police because of his inappropriate conduct in the street (e.g. the way they behaved in the street - maybe he was drunk and violent etc.).  In the example above, they are talking about the 'way' the investigation is carried out and they are not referring to her 'actually' carrying it out.  In other words, she assured that the investigation was carried out in the correct manner - that everyone involved played their part correctly and that all procedures that were followed.

Is the "assessing its effectiveness" a correct sentence or I had to use to assess like a verb??

When you assess something - you look at it, consider it, evaluate it and once you have completed that to the best of your ability and taking note of all the facts, you can then assess it and provide your findings.  The above use of assess is being used as a verb.  So Jessica is 'assessing' whether or not something has been effective by looking at everything, considering it and then evaluating it. She can then assess whether it was affective or not.  Her conclusion will be her 'assessment'.

"Due to clues arising from the documents, Jessica does something, solving the case".
Is "arising from" the correct noun to use in this sentence or also/only "coming out from" is fine?

Here 'clues arising' can mean 'showed themselves whilst assessing the document' - it's a more correct way of saying 'coming out from'.... other phrases could be "clues indicated in the document", "clues contained within the document", "clues hidden in the document" - they all have slightly different meaning ... "arising from the document" gives me a visual picture that the clues are there just waiting for someone with some experience (like Jessica) to find them if they look hard enough!

Thanks!