I am removing old hardwood and installing new engineered flooring. The old hardwood is 3/4" and the new engineered is 1/2". To match the height of the existing of adjacent tile, should i use 1/8" cork or 1/4". I realize the math, but want to make sure I am not creating another problem. Any suggestions?

Cork makes a great thickness shim.

We wouldn't know any better than you. Once the old floor is removed you will realize what's needed. Naildown, floating or gluedown engineered? Naildown or staple may not have enough penetration into the subfloor as the fastener specs are shorter for 1/2"