Is there a way to apply conditional formatting or grouping to reports?

You cannot set up conditional formatting in reports themselves, because they will reflect the conditional formatting of the sheet(s) they report against.

If you know how you'd like the results of your report to be grouped, you can get that done by setting the rules up for the formatting in your main sheet. If you need further assistance with this, then you might want to share some additional details.

Our organization has done a lot with this in terms of setting up conditional formatting in a sheet that is actually never even seen on the sheet itself (because it affects columns which are normally hidden on the sheet, but then appear on reports). We have also reported from two different reports, and have the conditional formatting between the two sheets "play nicely" with one another which gives the appearance of conditionally-formatted reports.

This report (which I have temporarily published) has the illusion of headings and grouped sections. However, the headings come from one sheet, and the non-headings actually come from another. I sort the report by start date, and set the dates of the heading rows such that they appear as headings for the items "underneath" them. The result here is essentially "grouped" rows, which are not actually grouped at all in their home sheets. In this specific example, units starting on or before each heading are grouped together. In fact, the order in the home sheets is more or less chaotic, but we don't care about that, we only care about the orderly reports.

Does this look like what you're trying to achieve? I can't really get any more detailed than that, but this example might prove to get the gears going for you.