Hi!
I temp vaginally. I first I was confused about whether I should turn the thermometer on first and them stick it in or whether to just put it in and then turn them on.
But I decided on turning it on with the tip slightly in and then sliding/guiding it in afterward. This works best for me. No cold and weird outside temps this way.
I use probe covers from Albertsons or by BD thermometers (they are little throw away plastic tip covers you can use each day). It keeps the process hygenic and easy and you get like 25 to 30 for like two bucks! Totally affordable! As for the last question...I stick it in as far as the skinny part will go.
Glad I could share!