When writing an informal letter, is it advisable to use the em rule at all?

In an informal letter, you have great deal of leeway as to what punctuation you may reasonably use. In part this is because multiple approaches to punctuation can accomplish the same thing, and in part it's because the whole point (so to speak) of punctuation is to make the sense of what you are trying to say clearer.

Let's take the two examples you bring up midway through your question:

(1) "I hope I will overcome my father's death — it won't be easy, though."

(2): "I've found a purse — it must be Kathy's."

In both sentences, what you call the em-rule serves to separate one semiautonomous idea from another—and I think it does a good job of it. But you have other options as well. For example, you might use a period in place of the em-rule:

(1) "I hope I will overcome my father's death. It won't be easy, though."

(2): "I've found a purse. It must be Kathy's."

This will give each thought maximum separation within a single paragraph. Or you can use a semicolon:

(1) "I hope I will overcome my father's death; it won't be easy, though."

(2): "I've found a purse; it must be Kathy's."

This brings the ideas closer together, with only a partial break between them. Or you could use parentheses:

(1) (1) "I hope I will overcome my father's death (it won't be easy, though)."

(2): "I've found a purse (it must be Kathy's)."

This makes the second idea seem more like an afterthought. Or you could recast the sentences slightly and use a comma in place of the em-rule:

(1) "I hope I will overcome my father's death, though it won't be easy."

(2): "I've found a purse, which must be Kathy's."

This brings the two ideas into significantly greater intimacy, as a sort of compound thought.

The punctuation you use is up to you. None of the examples above is wrong, though people may disagree about which ones are preferable and though different choices may signify different things to different people.

And finally, the notion that em-rules should be reserved exclusively for marking dialogue appears to have no authority behind it.