Dairy is fine for keto, just try to stick to low carb dairy such as cheese and heavy cream. Milk, because of the lactose content (it's a damn sugar folks) is not a great idea for maintaining ketosis. Just keep the carb content of your dairy in mind and count it towards daily totals.

Gum, being such a low glycemic index food (using the term loosely here), is probably not going to harm your ketosis (unless you chew a TON of it), but the artificial sweeteners often used in it, plus the soy lecithin, and all the rest, not a good option for other reasons. And for some folks it may keep their sweet taste trained and on ketosis you definitely want to curb that if only to limit cravings.