Barriers to Compassion

The Three Characteristics are teachings or vipassana objects for enlightenment.

The four Brahma Vihara are social attitudes rather than teachings for enlightenment.

Compassion is the wish for others to be free from suffering, including the wish that evil doers be free from the suffering that pushes them to do evil deeds.

Sympathetic joy is rejoicing in or appreciating the good actions of others, such as if a person saves the life of another person. Sympathetic joy is not something practised towards evil doers. Only three Brahma Vihara can be practised towards evil doers.

The noble path has three trainings therefore Brahma Vihara is part of the training in Right Thought & Right Conduct towards others, which is based in conventional reality rather than the ultimate reality of Emptiness.

A Buddha is perfect in both enlightened understanding & conventional behaviour.

What is subjected to 3 characteristics:

• what is conditioned - Dhammapada Verses 277, 278 and 279
• each link of Dependent Arising - (Jāti) Paccaya Sutta
• 5 argerates - Anatta Lakkhana Sutta, (Khandha) Sīla,vanta Sutta
• 6 sense bases, their respective objects and experiences - Pubba Sambodha Sutta 1 (Ajjhatta), Pubba Sambodha Sutta 2 (Bahiddha)

Generally you should try to understand the 3 Characteristics based on one of the above frameworks. Also the 3 Characteristics are not the end it self but means to an end, viz., because of impermanence, things are unsatisfactory, what is unsatisfactory should be lung to as me or mine.

The Brahmavihara also is conditioned. Therefore the result of such practice will be subjected to these characteristics, but this practice serves a different purpose. The purpose develop such qualities by breaking barriers on how you classify beings whatever the criteria. It severs a useful and benevolent purpose, hence is it not an empty practice.

Also the Equanimity is one of the Brahmavihara. Even when you wish someone well and it does not materialise be equanimous.

I think the main topic of the question is "sympathetic joy", not "compassion".

Muditā is translated for example here as:

muditā: altruistic or sympathetic joy, consisting of rejoicing at others' success and good fortune.

Wikipedia defines it here but without first-hand reference to scripture.

In the suttas which describe the four Brahmavaharas it's defined as "devoid of ill-will".

He dwells pervading one direction with a citta imbued with muditā, likewise the second, likewise the third, likewise the fourth; so above, below, transversely, everywhere and in every respect, he dwells pervading the entire world with a citta imbued with muditā, abundant, extensive, boundless, devoid of hostility, devoid of ill-will.

There's a long definition of the Brahmaviharas in Chapter IX of the Visuddhimagga:

1. Starting with loving-kindness, even towards a hostile person
2. Then (about 17 pages later) about developing compassion, even towards a robber about to be executed (but if you feel too hostile for compassion, try loving-kindness as previously described)

3. Then try to develop gladness (mudita), for which the advice is as follows:

   

4. Fourthly, equanimity.

From that I take it that it's a sympathetic joy: if you "feel" compassionate when someone else is sad, sympathetic joy is what you feel when, because, taking your cue from someone else being happy.

Another explanation is that the purpose of the practice is to make you devoid of hostility: devoid of jealousy, for example (e.g. "I'm happy for his success" rather than "I'm envious of his success").

Dhammadhatu's answer might be right as well. For example, some people might feel glad because of good or skillful actions. If you feel their gladness then you too are made glad as a result of good or skillful actions.

With practice, you can be content by reflecting on the good karma of others rather than joy or happiness. It is very close, if not the same, to praise. For example, if someone you're showing compassion to is now able to feel happy, then you can think "Now this person has the opportunity to practice dhamma since his problems are lesser. This is good". Maybe you can go further and help the person practice dhamma even for a second. This way, you don't cultivate thoughts that nurture attachment, nor do you appreciate any wrong joy or happiness in the other person you helped (such as thanha).

First of all you are veering dangerously to nihilism (annihilism). Seeing things as empty impermanent, not self and suffering is to allow us to detached ourselves to pleasures (including the pleasures of the bliss of Jhana). It doesn't mean that everything is meaningless or there are no consequences to our actions.

Emptiness in particularly used by Mahayana to mean that things are ultimately dependent originated. Things happen due to causes and conditions. It does not mean that it's all meaningless and does not matter. This is use to break your attachments to things you cling to as permanent. We should definitely not take this to mean that all actions are meaningless. If we do so we would quickly enter a deluded mindset.

Secondly Mutida or sympathetic joy is the antidote to jealousy and judgement. When we see someone who is better off than we are, we have a tendency to become jealous. Especially we have judged that person as being unworthy of happiness. On the other hand, if we feel they are worse off than we are, we might start to develop conceit, or even pity, and we become unable to rejoice in their good fortune. You can be happy for the limited happiness that they may have, whatever it maybe.

I want to further add that there is one powerful use of sympathetic joy, the joy of rejoicing in the merits of others, that is in their goodness. When we see others doing good things or making progress in the path, it is worth rejoicing, and in fact further aid and reinforce your own cultivation. Commonly you will see Buddhists exclaim 'Sadhu!' - meaning good or well said after seeing good deeds.