Can you still have high blood sugar if you are losing weight?

Of course. Someone could eat nothing but candy and sugary foods and still lose weight if their calorie intake was low enough. 

Yes depending on the cause of the high blood sugar 

Sure. Your blood glucose level is related to the amount of carbohydrates you consume as well as your body's individual ability to metabolize those foods and use then as fuel.

People with T2 Diabetes, Metabolic Syndrome, and Insulin Resistance all have difficulty using blood glucose efficiently.  The only way to really know if you have high blood sugar is to get tested and to know your A1C levels.

                          Yes

You're probably still eating foods which cause blood sugar spikes. If you're concerned, see a doctor

Yes - you can because high blood sugar has more than just one cause.

Age and heredity can cause high blood sugar also.  If age or heredity is the reason for high blood sugar, then dropping weight is not going to fix it.Also, you might not have lost enough weight to make a difference yet. 

You might still be eating too many carbohydrates.  (remember, controlling sugar levels is not just about cutting "sugar".  All carbohydrates become sugar in the body, so eating the wrong kinds of carbohydrates or too many carbohydrates can also cause problems with blood sugar levels - especially for someone who has been diagnosed as either diabetic or pre-diabetic.)  Consulting a dietitian might help.