How do I read the rhythms in this measure?

[mathandmusic]

"C" denotes common time (4/4) but there are more note values than beats indicated by the time signature.

Also what does that "6" represent?
Also in the next measure, what does that "quarter = quarter dotted" represent?

 



Tell me this is impossible and that Rach is cheating the notation system 

[j_menz]

"C" denotes common time (4/4) but there are more note values than beats indicated by the time signature.

Also what does that "6" represent?

The 6 means that you have six semi quavers in the time of 4. Just like triplet notation, but using semiquavers as the base unit.

Also in the next measure, what does that "quarter = quarter dotted" represent?

It is used to keep the pulse even following the change of time signature. Without seeing thefull bar, it is a little difficult to explain in operation.

EDIT: This page does a pretty good job: https://www.musicarrangers.com/star-theory/t15.htm

Tell me this is impossible and that Rach is cheating the notation system  

Nope, and nope.

[keypeg]

The first measure has four beats to the measure, like you said.  You can see four groups of notes joined together by a bar, except where there are rests.  The 6 means that there are 6 pulses in every, beat.  If you count your values, you'll see that each pulse must be a 16th note, since there are 6 16th notes in each group.  (one 1/8 being two 16ths of course). 

The second measure is in 12/8 time which often has the beats in triplets.  It probably means that each quarter note actually has the value of three eighth notes (dotted quarter) but they didn't want to put in all those dots.

[mathandmusic]

I see. Interesting use of 6's there.

[mathandmusic]

The 6 means that you have six semi quavers in the time of 4. Just like triplet notation, but using semiquavers as the base unit.

How would I count the group of 6-16th notes?

[j_menz]

How would I count the group of 6-16th notes?

Since there's no polyrhythm, I'd just go with 123456 123456 123456 123456.