with what u r going thru u need to be careful u don't overthink this. just spending time with him right now should be helping. if you can relax, not be sad for while it is OK now don't worry about lster. dont rush it u will find out soon enough. if he's not caring and respectful its not OK.
You have not been "dating" a whole week. You went out once last weekend. Duh. I recommend you not refer to him as your boyfriend yet lest you scare him off; you're thinking too far ahead too fast. If he continues asking you out consistently over the next 4--6 weeks, THEN you may think he's your boyfriend.