This is perfectly common in English, however the when should be a that. You are really combining "How often does it happen that X" and "Why does it happen that X". ("How often... when..." does not really make sense, after all.)
How often and why does it happen that packages get lost in the mail?
This works with other interrogatives, or a higher number of interrogatives, too — as long as you are combining sentences that only differ in the interrogative.
- When, where, and by whom were you robbed?
→ works, combines "when were you robbed" with "where were you robbed" and "by whom were you robbed", which all make sense.
- *When, where, and who robbed you?
→ doesn't work (that's what the asterisk means) because it combines "who robbed you" with "when robbed you" and "where robbed you", which are ungrammatical.
Here are a few example sentences from the Corpus of Contemporary American English:
- When and why did they die out?
- Where and when shall I meet you?
- Who or what is Gumby?
- Where, when and why do women take up horn?
(As a side note, there were a few other minor issues with your sentence that I fixed. The the is not justified because it makes it sound as if you were talking about the same packages being lost over and over again, while your intention is to ask about packages in general. The it can only be a this if you ask something like "How often and why does this happen?" and leave it at that. And the get can only be a got if you ask something like "How often did it happen that packages got lost".
Lastly, as StoneyB points out in his comment, you can lose the "does it happen that" part in favor of a simple "do". "How often and why do packages get lost?")