Learn English – The use of “actually” and “whatsoever”

Actually can be used in the present, but your second example doesn't make sense. In contexts like these, actually is generally used to contradict something said:

Person 1: You're a doctor, right?

Person 2: Actually, I'm a nurse.

Currently is used to indicate something that is happening now, and actually can be used like that too, so long as it is correcting/contradicting something.

In some cases, actually can be used to mean currently without being contradictory, but usually (in my experience) it is only used like that when the information will be coming as a surprise:

Person 1 (knows Person 2 as drug addict from high school): So, do you have a job?

Person 2: Actually, I'm a lawyer.

I use just very often in speech and casual writing. I tend to edit it out of more formal writing. I am a native American English speaker from Tampa and Boston.

To me, it can have both meanings, and I don't think that I have a preference. Your examples are all ambiguous to me. I think that the meaning is often clear from context (don't underestimate the power of context!), as in:

1) I'll be there soon. I just woke up. (recently)

2) I didn't call her. I just sent her an email. (merely)

3) I tried, but I just don't understand. (simply)

I don't see how changing tenses or aspects can help you, except that the future and simple present (always?) rule out the recently interpretation. But all of the following are equally ambiguous to me (sans disambiguating context).

4) I [just ate/am just eating/was just eating/have just eaten/etc.] an apple.

When I want to be crystal clear, I just replace just with merely, only, simply, or suchlike. I think that I less often change just to recently. I think I more often say just now to indicate that I mean recently.