How does one know that this is "...he will come" and not, ...he came?
"Que ele venha" is present, but in subjunctive (used to express doubts, etc).
So why not "I doubt that he comes"?
Present subjunctive is also used to express future actions (maybe that's why Duo has used that in the future) but what you quoted is also suitable. Languages are not literal translations. =)
I also think that is sounds better to say "I doubt (that) he is coming"
Why not using THAT he will come? It is proper English and even more important it corresponds to the Portuguese sentence.