Babies spit occasionally, and so long as the baby is thriving and growing normally and meeting her developmental milestones, we do not worry much about it. We call such babies "happy spitters" as they have regurgitation every now and then but are not irritated and they go back to feeding. This condition is known as Gastro-esophageal reflux (GER). GER is defined as the passage of the contents of the stomach into the lower esophagus (the esophagus is the tube the connects the throat to the stomach). It is considered a physiologic (i.e., not due to disease) occurrence in healthy infants, and results from the immaturity of the muscle sphincter that separates the lower esophagus from the stomach and that usually prevents backward return of the stomach content into the esophagus. In the majority of instances episodes of reflux are brief and do not extend above the lower esophagus.
GER is particularly common in infancy and manifests most frequently as episodes of regurgitation or vomiting. The majority of cases of GER are harmless and self-limiting: 50% of infants vomit at least once per day in the first 3 months of life, the peak of GER occurs at the age of 4 months with as much as 67% of four-month-old infants having reflux, then drops to 21% by 7-9 months of age.
As the weather has gotten colder, just make sure the room where you place your baby remains warm enough (around 25 degrees Celcius) but not too hot, as cold exposure may upset the baby's intestines and induce vomiting or spitting.
As for how you handle your breast milk when you are not breast feeding, we have already discussed this extensively in previous answers.