Well, I think I would go with everything said so far, for the DME
it is normal practice in our club to have the DME on, even in the circuit, on the basis that you never know when you might have to leave the circuit unexpectedly and may want/need to use it
this is fine, and as has been said, this should be in the checklist you are reading in the after start checks, possibly in a blanket "Navaids on and set" so make sure you read and do the checks properly to cover this.
For airborne checks, sit down in the evening and mentally run through the circuit from take off (or even before), climb out (think drift), when to turn crosswind,drift, level off, when to turn downwind, drift, when to call downwind, checks (I teach as suggested - carb heat on, checks with carb heat off at the end), then think if you made the downwind call as it can get missed in the correct place because of other traffic, then think about turning base and getting the aircraft set up, turn final and try and get the call in straight away (in a busy R/T environment it can even be worth calling before you are fully rolled out). Run through all this in your mind on the ground until you can do it over many times without missing bits and this will hopefully mean you then fly it the same way - also think about what to do when you cannot get calls in in the right place. At the end of the day though, if you are flying the aircraft OK, don't get too worked up about the odd minor error.
I also agree with Whopity, if your instructor thinks you are ready = you GO!