Q: Would sparsely repeated negative prompts have negative long-term outcomes?
A: Probably not, but more research would be needed to provide a definitive answer.
Norem & Chang (2002) argue that how negative (or positive) cognitions affect individuals depends on their predisposition. For example, a negative prompt given to someone with unrealistic optimism might call their attention to potential areas of improvement in their life otherwise overlooked, and lead to improved outcomes. Another perspective is self-affirmation theory, which suggests that healthy individuals would simply counterbalance negative prompts with positive thoughts - such as "there is much more going well in my life" or "others are doing even worse" - see also self-enhancement for a similar effect.
Another interesting suggestion comes from a classic experiment (Wilson et al, 1984), in which subjects were asked about their romantic relationships, and the mere act of making explicit their previously implicit attitudes changed those attitudes. This effect happens irrespective of valence - that is, asking to "Describe your spouse" potentially has the same effect as asking "What annoys you about your spouse?" - and is not affected by repetition - attitudes did not change significantly from being asked the same questions again in multiple sessions. Crucially however, changes in attitude did not result in outcome changes: There was no difference in breakup rate at followup. On the other hand, other studies (eg, Rubin & Mitchell, 1976; Veroff, Hatchett, & Douvan, 1992) found that such questions can affect marital satisfaction, and may accelerate its progression (strengthening or weakening it faster, again irrespective of valence).
Notes about domain specificity:
I am not aware of an experimental paradigm that could be used to test this specific question - ie, I don't know why any researcher would want to test if they can cause depression or other life problems in subjects, or what ethics review board would approve it. There are studies that look at long-term outcomes of positive prompts (eg, "What's going well in your life?"), but research in general suggests that positive and negative cognitions are independent (eg, Macleod & Moore, 2000), so we could not assume that negative prompts produce opposite reactions.
Furthermore, cognitions (both positive and negative) come in many flavours - referring to the past (eg, "What has gone wrong in your life?"), counterfactuals (eg, "What could have gone better in your life?"), future directed thinking (eg, "What will go wrong in your life?") also known as optimism/pessimism, and others. Unfortunately again, these often appear as different constructs with independent outcomes, so studies looking at any of these would not necessarily inform us about the type of questions you have in mind.
Priming is yet another related construct that does not usually look at long-term outcomes. In a typical affective priming paradigm for example, subjects may be primed with brief exposure to the words "failure" or "success", and then asked "How are things going in your life?" to see if they describe positive or negative aspects of their life more often. Priming effects are not believed to last very long - on the order of seconds (eg, Hermans, De Houwer & Eelen, 2001).