When drawing the demand and supply curves on a quantity/product space for an upwards sloping demand, assuming the two curves intersect, I noticed that the traditional consumer surplus region lies below the equilibrium price, and possibly even under the supply curve. What happens to consumer surplus in this situation? Does it become negative (as an integral definition would suggest), or do we simply say it defaults to zero?
Actually, neither demand for Veblen good nor for Giffen good is strictly increasing in price.
In case of Giffen good the demand actually looks as shown below in picture 1. The reason for this is that you can only increase demand for the Giffen good up until you consume your entire budget. Once the price gets higher then that you still get normal downward sloping demand.
Consequently if you would apply the definition of CS as the area below the demand and above equilibrium price (excluding the corner solution when Q=0) you would still get positive consumer surplus, as no matter what the equilibrium price is there still will be some finite positive area under the demand and above equilibrium price.
In case of Veblen good the demand actually looks like at the picture below. The reason for it looking like this is that the Veblen demand cannot start already at zero price, as for example if diamonds would cost 1 euro they could not be considered status symbol. Only at a certain equilibrium price when the good is not accessible to 'ordinary' people does the demand become upward sloping above the 'snob'line.
Hence, in this case as well the CS defined as area below demand curve and above equilibrium price will be positive.
This being said, I actually looked at all microeconomic textbooks I own, and I do own a lot of them both at undergraduate and graduate level, and none actually shows calculation of CS, or mentions how CS behaves for either of these cases even when they mention them. So to be honest I am not 100% confident that the CS idea would extend to these cases, but in principle blind application of CS definition to the above cases would still give you positive CS.