Sure. I don't even think it's necessary to wait until then. Sex should be safe at this time (not counting potential exposure/infection with viral STDs).
Is it safe to assume then that I can continue regular sexual activities with my girlfriend when I finish the Doxy?
Oral sex is safe sex -- not totally free of STD risk but much safer than unprotected vaginal or anal. Low risk for all STDs and zero risk for some. Had you asked before you treated yourself, I (and most STD experts) would have advised against it. That said, probably no harm will come.
Chances of infection were very low for any of the 3 STDs you mention. If you had been exposed, doxycycline would definitely prevent or abort chlamydia and syphilis, but not necessarily gonorrhea. When you started the doxy makes no significant difference.
Now that you have started, I would advise taking the doxy for a full week (100 mg twice daily for 7 days). Then move on without worry. With that treatment, the only one of these STDs that could show up later is gonorrhea, and if that happens you will know it from the symptoms (pus dripping from the penis, usually painful urination). If no such symptoms within a week of the last dose, you're home free. No testing of any kind should be done.