What are the implications of “Self-Reliance” by Emerson for religion, government, and property ownership?

Emerson's essay "Self-Reliance" is a passionate appeal to radical individualism. People have lost their way, Emerson argues; they pay so much reverence to institutions of Education, Government, and Religion that they can no longer think for themselves. This has led to a gradual decay of the very institutions the people revere, because no original thought is keeping them relevant. Society is moribund, chained to its past, and the only way to fix it is to cast off the past entirely.
"Whoso would be a man must be a nonconformist," says Emerson, for "nothing is at last sacred but the integrity of your own mind." If a person wishes to live—to truly live, and not merely play at living by following all the rules society dictates—then that person must be true to themselves in all things, at all times, regardless of what anyone else thinks. Emerson asks,

What have I to do with the sacredness of traditions, if I live wholly from within?

Traditions are a part of the social contract in which most people participate, so to reject traditions is to violate that contract. This has massive implications for society, and Emerson specifically rejects the validity of both religion and government, as well as property ownership.
With regard to religion, Emerson says:

No law can be sacred to me but that of my nature. . . . Good and bad are but names very readily transferable to that or this; the only right is what is after my constitution, the only wrong what is against it.

Emerson strongly believes that a person's inborn sense of right and wrong is a truer moral compass than any codified morality a religion may offer. It is the business of religions to instruct their adherents in how to think (that is the very meaning of "orthodoxy," i.e., "right-thinking"). Emerson therefore refutes religion—not because he does not believe in a higher power, but because he believes religion is an obstacle between the soul and the divine. Only without religion getting in the way can a person's spirit truly apprehend God:

The relations of the soul to the divine spirit are so pure, that it is profane to seek to interpose helps. . . . If, therefore, a man claims to know and speak of God, and carries you backward to the phraseology of some old mouldered nation in another country, in another world, believe him not.

Neither should people abase themselves before God in prayer in the hopes of getting something in return, for people are instruments of God's will, not petty clients. Prayer should be used to praise God, not beg favours from Him:

Prayer as a means to effect a private end is meanness and theft. It supposes dualism and not unity in nature and consciousness. As soon as the man is at one with God, he will not beg. He will then see prayer in all action.

If a person communes with God individually and does not use prayer to petition the Almighty for favours but rather to enhance their communication with God, then that person has no need of organized religion to dictate their morality. Therefore, Emerson argues,

As men's prayers are a disease of the will, so are their creeds a disease of the intellect.

Religion muddles the pure light of God's presence with rules, regulations, terminology, rituals, and jargon, all of which confuse people and make God seem distant instead of immanent. Emerson wants people to reject religion and thereby truly achieve the connection with the divine which religion purports to provide.
So much for religion. With regard to government, Emerson does not say anything in particular, but much of his essay may be understood as applying to government, or rather, the institution of "Government" in the abstract. It is Government which prescribes the boundaries of society, the laws which define it, and the consequences for breaking those laws. Government is necessarily a collective force, and it works on behalf of society as a collective, likewise. The "form and habit of government" is not a "disease" the way Emerson feels religion to be a disease, but the radical individual, the self-reliant person, cannot let government control their actions. There is no virtue in bowing to convention, even if that convention is codified as law, for as Emerson points out:

Society never advances. It recedes as fast on one side as it gains on the other. It undergoes continual changes; it is barbarous, it is civilized, it is christianized, it is rich, it is scientific; but this change is not amelioration. For every thing that is given, something is taken. Society acquires new arts, and loses old instincts.

By conforming to the collective, the individual loses their individuality. Society changes with the times, but it, in itself, cannot have any virtue; only the individual can be virtuous. Only the individual can achieve greatness and leave a lasting legacy.

An institution is the lengthened shadow of one man . . . and all history resolves itself very easily into the biography of a few stout and earnest persons.

Therefore, government should only be obeyed insofar as obeying its dictates is in alignment with the individual's own sense of virtue. If the law of the land goes against the individual's morality, the individual has a moral obligation to reject the law of the land.
"I will stand here for humanity," says Emerson, "and though I would make it kind, I would make it true."

Let us affront and reprimand the smooth mediocrity and squalid contentment of the times, and hurl in the face of custom, and trade, and office, the fact which is the upshot of all history, that there is a great responsible Thinker and Actor working wherever a man works; that a true man belongs to no other time or place, but is the centre of things.

Religion obscures the face of God from man's sight, and Government seeks to constrain man's nature in favour of the social collective. Both of these institutions restrict people's ability to be all they can be, to develop their potential to its fullest extent. They should both, therefore, be rejected—Religion outright, and Government as and when it may be necessary.
Finally, with regard to the ownership of property, Emerson has this to say:

The reliance on Property, including the reliance on governments which protect it, is the want of self-reliance. Men have looked away from themselves and at things so long, that they have come to esteem the religious, learned, and civil institutions as guards of property, and they deprecate assaults on these, because they feel them to be assaults on property. They measure their esteem of each other by what each has, and not by what each is.

Property is not an institution in the way of Religion or Government; it is not an abstract social concept. Property is comprised of real, material things. But these things are only "real" in the sense that they are physical; they can be felt, seen, measured. Because the physical reality of things is more readily apprehensible than the spiritual reality to which Emerson refers, people are dazzled by it and fail to see beyond what they hold in their hands. They will sacrifice many intangibles to protect these tangible things—their houses, their lands, their money—and they will willingly conform to the strictures of collective thinking if that is the price of security.
Benjamin Franklin famously said:

Those who would give up essential Liberty, to purchase a little temporary Safety, deserve neither Liberty nor Safety.

The context of his words was different to that in which Emerson is speaking, but Emerson is arguing much the same point. The worship of things has led to the degradation of the human soul, as bit by bit, people give up their individuality in order to secure the objects they love. Their individuality is the greatest treasure they could ever possess, and they are selling it for worthless things. Emerson argues that this is an appalling waste and furthermore that nobody can acquire any thing that is worth more than their own character:

That which a man is [he] does always by necessity acquire, and what the man acquires is living property, which does not wait the beck of rulers, or mobs, or revolutions, or fire, or storm, or bankruptcies, but perpetually renews itself wherever the man breathes.

If people give up their devotion to physical property, however, many of the laws society is built upon will cease to be necessary. Without the institutions of religion or government, and without even rules of property ownership, society as we commonly understand it would cease to exist. Emerson argues that this is a good thing, because society as we understand it no longer serves any purpose but to constrain the soul's true potential for growth, beauty, and greatness. We must jettison these institutions, as frightening as it may seem, for as Emerson concludes,

Nothing can bring you peace but yourself. Nothing can bring you peace but the triumph of principles.