I think the third condition is true conditional on Schanuel’s conjecture. Here is an example of using Schanuel’s conjecture to rule out the possibility that $e$ is the solution to $2^{2^u}-u=93$. I hope and expect that this (or some simplification) provides a template for ruling out all possible existential definitions of $e$. If the above equation holds with $u=e$, then consider the four pairs of expressions: $$w=\ln 2, \ e^w=2\\ x=1, \ e^x=e\\ y=e \ln 2, \ e^y=2^e\\ z=(2^e) \ln 2,\ e^z=2^{2^e}$$ We have five independent algebraic relationships among them: $$e^z - e^x = 93\\ e^w = 2\\ x = 1\\ y = w e^x\\ z = w e^y$$ By Schanuel’s conjecture there must be a rational linear dependence among the first four variables: $$a w + b x + c y + d z = 0$$ and dividing by $\ln 2$ gives $$a + b/w + c e^x + d e^y = 0$$ So the three pairs of expressions with $w,x,y$ have four independent algebraic relationships between them: $$a + b/w + c e^x + d e^y = 0\\ e^w = 2\\ x = 1\\ y = w e^x$$ By Schanuel’s conjecture there must be a rational linear dependence: $$p w + q x + r y = 0$$ and dividing by $\ln 2$ gives $$p + q/w + r e^x = 0$$ So the two pairs of expressions with $w,x$ have three independent algebraic relationships: $$p + q/w + r e^x = 0\\ e^w = 2\\ x = 1$$ By Schanuel’s conjecture there must be a rational linear dependence $$s w + t x = 0$$ This would mean $\ln 2$ is rational, which is impossible, so the original equation does not hold for $u=e$.