No they do not contain Diethylene Glycol. They contain Propylene Glycol which is the "safe antifreeze" ingredient pets can drink without harm, concert fog, and the way hospitals deliver inhaled medications, even to lung transplant patients.
Re how much to vape - there is no way for anyone to know that but you. We do not know the size of your mouth or lungs, the efficiency of your atomizer or cartomizer or any other information that would allow us to guess. When I first started vaping I just tried to vape as often and for as many drags as I used to smoke. Later many people start vaping more often for fewer drags but with some sessions that have a lot of drags - i.e. just vaping for however much you need. Sometimes I need 2 drags. Sometimes I need 20+ drags (like in the morning).
Many people vape 1.5 to 2.5 ml per day. Nicotine in vapor is not absorbed the same an in cigarettes. There is not direct comparison. If you get too much nicotine you will probably get one of more of the following symptoms: headache, rapid or irregular heartbeat, cold fingers/toes, hazy or ill feeling (but when you first switch to vaping you may get that hazy/ill feeling anyway as the 4000 chemicals in cigarettes leave your body).
Drink plenty of fluids - vaping can be drying. And it doesn't hurt to take a multivitamin for the first week.
If your mouth, tongue, throat or lungs get sore - that's usually nicotine skin irritation. Get juices with some VG in them. 20-50% VG is nice in atomizers, 20-30% for tanks, 10-20% for cartomizers. VG helps buffer the irritation though it dulls flavor and TH (but 10-30% VG still leaves plenty of flavor and TH)