Your paper mulch would look like it got shot with a shot gun. I wouldn't even attempt to grow carrots in plastic or paper mulch.
Plant them directly in the soil, cover with row cover or some sort of landscape fabric to keep moist to germinate, flame weed on about day 5/6 and then hand weed once or twice until they canopy and forget about them.

Jay

and hand planting hundreds of tiny seeds would be nightmarish and take hours to do.

we usually plant carrots with radishes so we know where the rows are and when harvesting the radishes you get most of the weeding between plants done as well as most thinning. Hoeing early and regularly as needed keeps the weeds under control.

I do close together rows with radishes too- several passes with the seeder in the bed, and then lightly hoe once or twice by just pulling the hoe down the space between rows. Carrots must be handweeded and personally I find straw mulch on carrots, beets and lettuce to be annoying when picking so I just straw mulch legumes and most everything else gets plastic mulch. Because of the handweeding carrots are a more labor intensive crop but at least they don't require much soil amendments. And now my mom weeds and stuff a couple hours per week for me.

Thanks for your help!