is my understanding correct?
Not really. Functional programming is more a set of features originally inspired by lambda calculus than a foundation for an entire language design. Some purely functional languages might come close but I suspect they'd still have to have some rules that don't relate directly to lambda calculus. And of course not all languages have the features required for functional programming.
are there any models other than lambda calculus?
If by that you mean feature sets inspired by other branches of mathematics, then absolutely, without a doubt. Basic features implemented in most languages are tied directly to all manner of specific branches of math. But not all features necessarily are. Object Oriented Programming, for instance, is a concept meant to help programmers structure and model code more intuitively while avoiding some of the pitfalls people tend to run into in procedural programming. If it has a direct branch of mathematics analogue, I've never seen that pointed out anywhere.
if the languages shares the same model, can we use that to convert a
source code from one language to the other?
I think the closest thing to a common link you could find between all languages is that ultimately they all have to boil down to instructions that can somehow be broken down and executed in machine code terms on a given system (and machine code varies depending on architecture too). The complexity inherent in making a universal programming translator would be both insanely hard, if possible, but also massively fail to produce legible translated code between languages without massaging and special libraries that would make it fail the 'universal' test. I wouldn't be shocked if someone had proven this feat to be technically impossible.