When you say that you doubt something is true, it usually means that you think it isn't but you can't say that with 100% certainty. If a person says that they are doubting or starting to doubt something, it means that they are having doubts about something, but haven't decided yet. The doubts are stll circulating in their minds.

I doubt she is telling the truth is correct. If you believed previously that she was telling the truth and now something has happened that’s made you change your mind, you can say: I am now doubting whether she was telling the truth.

I don't think it's correct to use the gerund - ing in that statement. You can say I doubt or I'm in doubt. When you use I'm doubting, it's doesn't really sound right.

-1- Leo gave an excellent answer. -2- Practical English Usage by Swan is an excellent reference for questions at this level. -3- Stative verbs don’t usually take a progressive form. Ex: I own a car. [not - I am owning a car.] Some verbs such as “to have” have stative and dynamic meanings. Ex: I have a car. [= I own a car. (stative)] I am having breakfast. [= I am eating breakfast. (dynamic)] -4- There are now multiple Englishes independent of Britain and the US. What used to be called “errors” are now accepted as regional differences. A Singapore businessman can communicate with European and Asian business partners in Singapore English. In my experience as an engineer, my Indian coworkers (in India) spoke and wrote Indian English. One difference was treating stative verbs as dynamic (Ex: “I am knowing.”)

Hello! This is an example of a complex sentence that takes place in present. Probably, some girl is telling a story and you share your doubts with someone on that particular story. In the independent clause (I doubt) the Present is used in Simple tense as the verb doubt is a stative, a non-continious verb. In the dependent clause PrC is used as the action is still in process (she is telling a story now). If you put PrS (she tells the truth) that means she lies all the time. Have a great day!