Does your LH stay high when you are pregnant?

No. The "lh surge" precipitates ovulation, but once the oocyte is released (you ovulate), LH drops until the next cycle (which won't occur again for 9 months if the oocyte is fertilized and pregnancy occurs).
No. You will not produce significant amounts of lh during pregnancy. However, in rare instances the HCG (pregnancy hormone) may cross react with the lh test and give you a false positive lh test. This is very unusual. The most common reason for persistent lh test is pcos in which a women produces chronically high lh levels.