AH is proven to significantly increase pregnancy rates for women over 38 so it should be done routinely if you are 38 or older. If you are younger than that, it depends on the embryos. A good embryologist normally makes that decision before the transfer. It is more frequent for women with a low ovarian reserve. Sometimes, it will be done empirically for women with idiopathic infertility and previous IVF failures with good-looking embryos but it does not seem to make a huge difference on success rates. Sometimes, doctors will want to do it if there is a smaller than expected number of embryos but If it is a frozen embryo transfer, AH is always done as the zona pellucida reacts negatively to being frozen (or thawed).
Some patients get a little scared when we explain to them what exactly AH is but I can assure you that it is a lot less impressive than it sounds.
As far as pregnancy tests, I would discourage you from taking them. When doing IVF, we know that embryos have lived in your uterus for at least a little while. It is very common for HCG levels to initally rise but stop fairly quickly so a pregnancy from IVF is always only confirmed with 2 tests properly doubling 48hours apart, at least 14 days from the retrieval. You should also keep that in mind if the first test is positive. The disapointment when it does not increase at the seciond test is a horrible thing to experience and is so much worse when the patients where ill-prepared for that eventuality. So, in short, if you choose to test at home anyway, make sure ou fully understand that the results do not provide any conclusive evidence of an actual healthy pregnancy.
Good luck!