Dear Li-
With my first pregnancy, which occurred in September 2006 when I was 28 YO, I do not recall being tested for immunizations; however, there were a ton of blood draws in the first 12 weeks so it is possible that the OB ordered something like that and I never realized it.
With this pregnancy (I am now 32 weeks and due in September), NJ (where my OB and hospital both are) changed the rules and regulations with regards to HIV/Aids testing. You always have the right to refuse, but then they would have stuck the baby when born so I consented to the testing. They took an extra vial of blood at 8 weeks to run and again recently at 24 weeks I believe.
I have ALOT more experience with genetic testing. My husband's sister has Downs so even though I was not a "critical" age with either pregnancy, we obtained a referal for Maternal Fetal Medicine both times. They are associated with Valley Hospital and were WONDERFUL. We waited to get counseling until after we were preggers but certain things like the family genetic tree (i.e. nationalities back to great grandparents or further if possible on all sides), genetic testing for my hubby to see if the Downs was due to a "tag" or "tail" attached to a gene that could be replicated by his body and even my genetic profile could have all been done ahead of time. I do not know if my insurance would have covered it though. It was not an issue once I was pregnant.
In addition, there is certain "genetic" testing that can only be done once you are preggers. Things like the nucleotranslucey (sp??), using an ultrasound to identify abnormalities or potential problems with "typically" affected bady parts like the nose, sternum, heart, etc.
If you have questions or concerns, I would talk with your OB. I had a pre-pregnancy "screening" for overall health and whatnot and they should be able to guide you through the process better than any of us here at Mamasource could.
Best of luck.
~C.