If it is a 35% solution, it means 35g in 100g of water.
No it doesn't. 35 g HCl in 100 g of water is 35/135 = 26%. It means 35 g in 65 g of water. 35% of the mass of the solution is HCl, the other 65% is water.
Since density of water is 1. 100g=100ml.
Hence 10ml of the solution contains 3.5g of HCl.
So number of moles = 3.5/36.45=0.096 moles of HCl in 10ml of water.
No, this is wrong. You have ignored the density of the solution (1.174 g/mL).
10.00 mL of solution has a mass of 11.74 g, 35% of which is HCl.