The first ability is weirdly worded and atypical for white.

Mmm, white is the army colour; so an ability that triggers on having an army of is reasonable. Children as a tribe is... kinda odd, but sure, ok. Here we have a child lord.

If anything, I'd question the second ability - it might do nothing; it might be affinity-for-artifact-child-army-doom.

Those are some powerful children. They can go toe-to-toe with Imperial Lancers, presuming there are no dinosaurs around (though, admittedly, they lose fights to Sanctuary Cats.)

I'd use 0/1s. But I figure you already thought about that and decided 1/1 was fine.

Yes, I do regret giving white a draw engine.

I never made anything about Eluim's setting clear. Well, it should be clear that Eluim is an artifact set, so a card interacting with artifacts makes sense. Lady Reomna created a device Lady Reomna's Petriwombto rapidly create children, attempting to develop one whose spark would ignite under forced duress in the petriwomb Abort Test. Lady Reomna was the wife of Reyles (the general of Eluim's military). Lady Reomna's petriwomb did work, but the planeswalker whose spark ignited Ialma, the Curious also destroyed the petriwomb, killing Lady Reomna.

Anyway, that's why Lady Reomna makes child tokens.

The most common way the ability would be worded is with an intervening if-clause and the trigger condition in the beginning:

> At the beginning of your upkeep, if you control seven or more Children, draw a card.

Without intervening if-clause (you know, like what you have right now :) ) is also okay - it's a slight functional difference, but generally not an issue. (FYI: With the if-clause in the end you don't need the seven Children until the resolution of the ability e. g. you could put this on the stack and activate the ability in response to create the last Child - with an intervening if-clause you better have the seven Children before the upkeep even begins!)

More relevant though is that unless you have a good reason (usually to avoid ambiguity or something IDR) you put the trigger condition in the beginning - that way a player can parse the condition immediately and skip the rest if the condition is irrelevant to them (right now). It also helps to have the trigger signal word as early as possible in the ability ("When"/"Whenever"/"At" as the first word of the ability is great if all you want to know is whether the ability is affected by Panharmonicon).

Has it been mentioned that Child is a terrible idea for a creature type?

I've commited far too much to the creature type at this point.