I think sedge too. Pull one and look at the main stem. Is is triangular? Growing leaves in roughly three different directions? (grasses grow leaves opposite one another, sedges grow them roughly 120 degrees from each other, hence the triangular stems.) Are the leaves shiny and waxy? If so it is almost without a doubt nutsedge. Put more simply, if the stem is triangular it is a sedge, and not a grass, and the only thing you need to figure out is what sedge, exactly, it is. But still if the stem is triangular and it is growing in your lawn it is probably some variety of nutsedge. Which is a problem, because these weeds grow underground storage systems (nuts - hence the name) to help them rejuvenate after you pull them. So to get rid of them you have to either pull them like every other day for a year, or apply a product like Sedgehammer. MSMA also has an effect on them, but it is relatively mild and not nearly as effective as the more expensive and harder to find sedgehammer.

I presumed it wasn't the grass because I've been in this location for 4 years and have never seen it in the lawn before. I used an herbicide in April this year that may have upset the balance of lawn weed infestation in favor of this new intruder. If this is indeed little barley, it should die off next month, correct? Something like foxtail barley would live on until the end of summer? I ask because I won't bother finding an herbicide for this if it will be dead in a few weeks. Also, I've read to apply pre-emergent for little barley in September, as opposed to the usual February application (for crabgrass). I'd appreciate any knowledge you could share on this.

That looks like Ailanthus altissima, tree of heaven, a very weedy and stinky tree. Here is a link that might be useful: Ailanthus altissima, tree of heaven

In your photo there are LOTS of little seedlings, which means there must have been lots of seeds in your lawn or in the compost you put down. You didn't mention the lawn having lots of weeds previously so I wonder about the compost. Where did it come from? Your options are limited while the new grass is just getting started since many broadleaf herbicides can damage the grass seedlings too. If you wait until the grass is established at least 8 weeks, then you may be able to use herbicides. Check the label of any product before using it on new lawns. I agree with kimmsr that correcting soil problems is the way to manage moss in a lawn. Moss is not a competitive plant, as you can see by the very limited root system it has. It grows in lawns where the grass is already doing poorly. This can be cause by lack of soil fertility (the most common cause in my experience), too much shade, heavy soils that drain poorly, and excess acidity (pH below about 5.2). Lack of organic matter isn't normally limiting for grass, but adding organic matter can improve drainage if that is the problem. Have you had a soil test done? You mention that the soil was determined to be acidic. What was the pH?