Given a simply connected smooth projective variety (hence a compact Kähler manifold) with singular cohomology generated in even degrees, do we know that there is a Morse function on it such that all its Morse indices are even? 



Kähler manifold with even-only singular cohomology
Assuming that by `cohomology' you mean integer coefficients, there is a general result saying what you want for simply connected manifolds of dimension $5$ or more, without the Kähler condition. According to Smale (Generalized Poincare’s conjecture in dimensions greater than four, Ann. Math. 74, No, 2, 391-406 (1961)) for $n > 5$ and Barden (Simply connected five-manifolds, Ann, Math, 82,365-385 (1965)) if $n = 5$ a simply connected $n$-manifold has a handle decomposition (ie Morse function) with the fewest handles that you need for a chain complex representing the homology. 


In your example, having only even dimensional cohomology means the same for homology, and hence there's no torsion. Thus there is a chain complex with only even index generators and so, for complex dimension at least $3$, by Smale a Morse function with the same property. 


The case of complex surfaces is not yet established, although there are many concrete examples where there are decompositions with only even index handles. (For example, hypersurfaces in ${\mathbb C}P^3$, elliptic surfaces without multiple fibers...). See the book of Gompf-Stipsicz for some of this. A more recent striking example, due to Akbulut, is the Dolgachev surface, equal to a two-fold log transform of orders $2$ and $3$ on an elliptic surface $E(1)$. This had been previously suggested as a counterexample to your statement. 


