COUNTIFS ignoring rows based on values in row - excel

Question

I have been struggling with these two problems for the better part of the day. Please see this link to an example: https://docs.google.com/spreadsheets/d/1odU7oqc-WviLzQcghkbYbQG-4DC70-Q_hszb22Chljw/edit?usp=sharing
This is a simplified document I use to see which lecturer (name) has has added which of the 5 documents (on the right) to an online learning environment (one row for each course). The courses can be given in semester 1, 2, or in both. I want to use this document to automatically highlight a name if any of the five boxes on the right is empty if the semester I want to check (Cell D1) is similar to Column E. Furthermore, only the first time a name appears in red, should the e-mail appear so as well.
For this, I of course use conditional formatting (please see those formulas, also pasted in column M (range column C and D) and N (Range column D). N is the white conditional formatting overruling M, in which a N="TRUE" means it is whitened out).
Example 1 (Ans) shows how I intend the programme to work. The first instance shows shows both the name and e-mail in red, later instances do too unless the semester is not similar or all boxes on the right are ticked.
Example 2 (Bertrand) shows the first problem. If the first instance of a red name and e-mail occurs in a semester 'both', the second time the name should be red the e-mail does so as well. This keep occurring until a semester is similar to Cell D1, after which it is normal again.
Example 3 (Carel) shows the second problem. If the first instance in which a certain lecturer occurs all boxes on the right are ticked, the second time he occurs but does not tick all boxes his name shows in red but his e-mail doesn't.
I think problem 1 (example 2) is solved if I can get 'both' to somehow mean both 1 and 2, but I don't know how.
I think problem 2 (example 3) is solved if I can get COUNTIFS to ignore the rows is which all five boxes are ticked.
However, I have no clue how to do this. Is anyone able to help me with either (but ideally both) problems?
Thank you in advance!

Answer 1

I'm not entirely sure, but I think I've understood what you need.
For the first part (where a lecturer has less than all 5 subjects containing 'x' AND where E is either the value of cell D1 or the value of 'both'), try this:
Delete both of your existing conditional formatting rules.
Create a new conditional formatting rule for range = C4:C21 (or perhaps further down your sheet to row 977), with a custom formula of:
=AND(COUNTIF($G4:$K4,"x")<5,or($E4=$D$1,$E4="both"))
The style for the rule should have a red background.
For the second part (updated), this is a bit tricker with conditional formatting but I can offer a solution determined by the value of cells, like you had.
Delete the contents of cells M4:N21 (21 being where you have a formula in the furthest cell).
In cell M4, add the following formula:
=arrayformula(if(iferror(if(ifs(E4:E=$D$1,1,E4:E="both",1)+if(G4:G&H4:H&I4:I&J4:J&K4:K <>"xxxxx",1,)=2,D4:D,),)<>"",countifs(iferror(if(ifs(E4:E=$D$1,1,E4:E="both",1)+if(G4:G&H4:H&I4:I&J4:J&K4:K <>"xxxxx",1,)=2,D4:D,),),iferror(if(ifs(E4:E=$D$1,1,E4:E="both",1)+if(G4:G&H4:H&I4:I&J4:J&K4:K <>"xxxxx",1,)=2,D4:D,),),row($E4:$E),"<="&row($E4:$E))=1,))
It might seem a little complex, but it's done this way to use the functions that are compatible with ARRAYFORMULA so that it will keep working down the sheet as far as you have values in rows.
Now create a new conditional formatting rule for range = D4:D21 (or perhaps further down your sheet to row 977), with a custom formula of:
=$M4=true
The style for the rule should have a red background.