No, the semi-implicit or symplectic Euler method applies to Hamiltonian systems, not general stiff problems. And when you use it, go one step further and adapt the initial velocity so that the method becomes without further changes the leapfrog Verlet method.
Yes, there is an implicit Euler method. You will have to solve an equation system for the slope, for instance with Newton`s method. The implicit method with one Newton step is the simplest Rosenbrock method.
In general it makes no sense to use the order 1 Euler method when with essentially the same effort you could implement the order 2 midpoint method or the order 2 trapezoidal method.
And if the machinery is in place to deal with the equation system of implicit methods, you could just as well consider higher order methods, see for instance Butcher`s tutorial slides for his book, esp. the third set.