Can one pray for something "bad" to happen to yourself?

Short Answer

As a child of God who is right in asking God for joy and fulfillment, do not ask God to remove a desire unless you simultaneously ask for a better one, because God is most glorified when we are satisfied in Him.

General principles

John 14:13-14 has been abused by many not-so-Christian groups so we should be very careful to understand what the writer himself (Apostle John) meant by it. There is already a fine answer about it, so I will not repeat it here other than saying that it hinges on what "asking in Jesus name" means:

So when we pray "in Jesus' name" we are praying "according to who Jesus is", that is to say according to his nature as the revelation of God; the salvation of his people; the way, the truth and the life; the gate of salvation. We are praying according to who Jesus most fundamentally is. To pray in Jesus' name is to conform ourselves to that identity.

Another aspect to consider is who is praying: is the person someone who has made Jesus his/her Lord? Does the person want the "gift" or the "Giver"? Does the person consider how the request is aligned with the purpose of the Kingdom, one of the utmost reasons why Jesus came to earth? Even Santa Claus would not give a child a gift that would harm the child, would he?

About "gift of salvation", some Christian groups teach that after we have joined the Kingdom of God (because Jesus has redeemed us) we are now eligible for the blessings that pre-fall Adam forfeited AND the blessings of the Abrahamic covenant because we have been grafted into the olive tree (Rom 11:11-24). This includes receiving answers to our prayers made in Jesus name as long as they are aligned with Jesus's own heart.

Proper way of relinquishing

In my opinion, there are 2 ways of relinquishing a desire. One is the Buddhist way: simply kill it, because increasing desire is increasing "self" which makes one further away from obtaining Nirvana, conceived as losing one's self (a drop of water) into the vast ocean of "bliss" (but the wrong kind of bliss, according to Christianity).

The other way is the Saints's way, illustrated by Ken Graham's St. Bridgid example. She relinquished beauty for a better desire, similar to a monk's taking a vow of poverty. Buddhist poverty is very different than a Christian saint's poverty. Right desire for a Christian is to align ourselves with what Jesus desires, not to kill it. The Saints desire the growth of the Kingdom which is established on earth in many ways, which explain why Saints's lives are radically different from one another. Some worked in the world (like the Jesuits), some retreat into the desert to pray for believers, some build hospitals, some minister to the despised like Mother Teresa, etc. But all Saints relinquish something personal for a greater good that pleases God. All Saints don't just ask for something "bad"; they simultaneously ask for something greater to replace the "bad" with.

Application and responses to your questions

Applying the above to the situation you were talking about requires us to know what the event is about. Rather than reconciling to "futility", the person should instead ask himself how God would have used the event to bless him and others participating in the event. God works in mysterious way, opening and closing the doors, which should NOT discourage us since our trust is in the Giver. So rather than NOT asking, the person should instead ask God to reveal His purpose for him in relation to the event.

Rather than asking your parents to have some chocolate, you give up and just never ask, and never eat chocolate, and accept that as normality. Where is the benefit in this?

I agree with you; there is no benefit in this. If a parent does this to a child, the parent has harmed the child psychologically! I personally feel that behind the invitation of John 14:13-14 is a revelation from God to His Kingdom people, who are experiencing hardship in the world of suffering, to NOT give up asking. Assuming the desire for chocolate is healthy (i.e. not driven by obsession / greediness / substitute for nutritious food), God wants us to keep asking and to praise God when He delivers. Instead of giving us chocolate, God may want to give us something else that is better for us in the long run, such as chocolate ice cream after we finish a work of mercy to others.

losing an interest is something I consider "bad" if it is not replaced with something else meaningful.

I totally agree. So unless relinquishing desire for the event is for a "greater good" (like Ken Graham's St. Bridgid example), then it is suspect. In St. Brigid's example, her request to make herself look ugly was to pave the way for something far more meaningful: being a consecrated virgin for Jesus.

Would it be ok for a child tell Santa Claus "Don't bother with this and that toy, I won't be getting it anyway." Would Santa Claus be ok with that, or would he be miffed? And how would that serve Santa other than his Elves don't have to make the toy? God has no material costs incurred when fulfilling a prayer, so how can God benefit from "Pack it in, God. I don't want it any longer!", and how can this serve His kingdom if no such intent is implied in prayer? Lastly, can the rejection (read: giving up) of anything purely due to its unattainability be a token of mercy to the person praying?

Santa Claus / Jesus would be sad. There are a few things wrong with that view:

1. That attitude seems like a mask for accepting destiny, which is not Christian. Serving in the Kingdom is instead a joyful participation with fellow believers following Jesus to victory! If the thing to be given up is morally wrong, that's a different story. But giving up a potential blessing to me is contrary to being a child of God. God may say no to our request (when it's not aligned with his purpose), but He would give us other avenues to bless us. So closing door would open our desire for another opportunity, not leaving us unfulfilled.

2. Yes, because of God's unlimited wealth of blessings, God wouldn't "incur material costs" when fulfilling prayer, but it's God's will to share his blessings with us, as is clear in the parables of Jesus and elsewhere in the Bible. If the blessings don't come in the form we expect then probably we need to adjust our expectation, but ceasing to ask is not the way.

3. About "token of mercy" it's possible if God considers the request to be harmful for us, but that implies that we ask Him first. As a matter of courtesy, we shouldn't ask God with a stoic expectation / stiff upper lip that it wouldn't matter whether God gives it or not. That's a beggar's mentality, not suitable for a child of God ! What parent likes a child who asks with that attitude? At best, God would be sad. We should instead be like Jacob who asks for what he desires and not give up, or like David (when his & Bathsheba's kid was dying) until he found out God says "no".

It seems like God opened a prayer trap, as it appears that it is easier to just pray for an increasingly empty life and receive it, rather than have it fulfilled with the joys God can give through his mercy.

I agree that praying for empty life (and receive it as though it's God's answer) is WRONG. Instead, joy should be primary. For an antidote of that feeling please check out What is Christian Hedonism? by John Piper who says: "Over the years the name that I have given to my understanding of the massive role joy plays not only in the Christian life, but in all of creation and God’s purposes in it — is Christian Hedonism. And the shortest description of Christian Hedonism is God is most glorified in us when we are most satisfied in him."