Why don't Presbyterians have altars? Why don't Presbyterians have altars in their churches or in their homes? Who administers the sacraments, especially communion, then without an altar? Do children help out at all? In other churches, children can be selected as 'altar-servers'.
(Although I can't speak for all Presbyterians, as we are diverse bunch, I think that what I say here is representative of the mainstream. I welcome correction if I am wrong about this.)

When Communion is served, we do use a table of some kind. The table recalls the Last Supper, at which the practice of Communion was instituted; Matthew 26:20, Mark 14:18, Luke 22:14 and John 13:23 all say that Jesus and the disciples were gathered at table. In any case, it is practical to have a flat surface for everything else to rest on.

It is typical for the table to be an actual table (wooden, four legs at the corners, etc.) rather than an altar (big block of stone). This is because the Reformed tradition does not view Communion as a sacrificial event, and does not have the Catholic concept of a sacrificing priesthood. An altar is a place for offering sacrifices but a table is a place for having dinner. Also, there is a general aesthetic sense that churches ought not to be too ornamented, so some may prefer a simpler piece of wooden furniture to an elaborately carved stone altar. I have personally seen quite a bit of variation in the level of decoration for the table.

Moreover, the Presbyterian tradition has tended to offer Communion less frequently than some other denominations. So the table may not even be a permanent fixture, but is brought out whenever it is needed. In contrast, the preaching of the Word is ubiquitous, so in some church buildings you will see a central pulpit where you might expect a table or altar. In my present church, we do have a permanent table, where the Communion vessels are always shown - this is meant to be a reminder that we should be ready for Jesus to arrive at any moment. In other churches you might see candlesticks, a cross, a Bible, and so on.

You also ask who administers the sacrament, and whether children may be involved as helpers. We do have a person who does this, a "minister", who has been ordained and who (as a result of training followed by church approval) is considered capable of the job. Church law may allow other people to act in place of the regular minister, under certain circumstances. The minister may be helped by other people during the course of the service - holding plates of bread and so forth. Compared to high-liturgical denominations, the Presbyterians have rather less apparatus to deal with (incense, bells, different kinds of cloth, etc.) and so there is correspondingly less of a role for the helpers. They also don't have special clothes to wear.

Regarding children, I would think that anyone assisting ought at least to be baptized - even if they do not receive the sacrament themselves, helpers are still participating in the event. They should also be physically able to do what is asked. So I think anyone who could receive communion could also assist, including children in principle, whereas unbaptized people of any age would not do so. Actually choosing the helpers would be the responsibility of the session (the governing committee of the local church) together with the minister, ultimately subject to the judgement of the presbytery and higher bodies. Again, given the nature of the role, there's not much need for more than about half a minute of training, so we wouldn't tend to have a fixed body of designated helpers - at my present church, the session chooses people from the congregation in general.