Cardio and/or strength

This is discussed greatly on these forums of losing fat and building muscle. So I would recommend you doing a little research for further understanding. But in a nutshell, if you are trying to lose weight and you are cutting calories by diet and exercise (running for your example) for a goal of losing weight, then the only time you will appreciably build muscle is if you fit the criteria laid out by what Lyle McDonald discuss in this link http://www.bodyrecomposition.com/fat-loss/adding-muscle-while-losing-fat-qa.html/
That being said, outside of those circumstances, you still maintain what you have or minimize loss by lifting to stimulate muscle (also assuming adequate protein intake). You can technically do both at the same time, but to a MUCH SLOWER extent by recomping (again, something discussed many times in these forums, so do your research). The more optimal approach, for quicker results, is to do cut and bulk cycles opposed to recomping, but that is a personal choice.

Yes you can do both on the same day. If doing at the same workout period, do lifting first, then run. If you run in the morning and lift in the evening, then just take into account your body's ability to recover to be able to lift (which I see as more important in the grand scheme of overall weight loss and health)...but that's me and you might be different in that you might love to run, and in that context, that's what keeps "YOU" healthy and active.