Haplogroup Q-YP1004

I'm Haplogroup Q-YP1004. This haplogroup starts in the Ukraine and my ancestors are Ashkenazi around 800 and some tests show I'm also Khazar. I know there is debate on the Ashkenazi and the Khazars. Would someone shed some light on the newest finding? Second
how would I find my ancestors past my 6th great grandfather if surnames were not added until between the 800's and the 1500's. Surname is Lake. Ukraine to Russia to Poland to England (1720 birth of my 6th GF) to US in mid-1700's)