Don't worry yourself girl, you're fine. I didn't have morning sickness that early on, it came a little later, and was real mild, except when i got headaches... and you may be lucky to just not get it at all... i didn't 'feel' pregnant until about 8-9 weeks.. every pregnancy is different. just because it's what usually happens, doesn't mean it has to in order for everything to be okay. (: try to relax.
A doppler wouldn't be able to detect the heartbeat until about 9-10 week at the VERY earliest. As for not having morning sickness don't worry most people don't start getting it until about 6-8 weeks. I remember I wanted to get it so bad too and then I got it and wanted it to go away! Constipation comes and goes so don't be concerned with that. But it sounds to me like your pregnancy is going well! :)