Hate to sound like a downer, but I think you might be seeing a sucker from the rootstock of one of your plants. Most roses sold in the US are grafted onto a rose called Dr. Huey, which is a once-blooming climber. You can tell by digging down and finding the base of the new growth. If it is a sucker from rootstock, the growth will come from below the graft union (see the photo below). That needs to be scraped off, otherwise, you will have all rootstock and the variety that it is growing from will die off.

It could also be a runner from your Flower Carpet or your McCartney Rose, but you would still have to dig down or wait for it to bloom to make sure.

Roses don't usually reseed readily. Their seeds are encased in a seed pod, called a rose hip, and it takes a bit of effort to get the seeds out and prepare the seeds for germination.

Here is a link that might be useful: Anatomy of a Bareroot from Regan Nursery

Hmm, I guess I'll dig down and check (goodbye junipers!). It's definitely not from the carpet rose, too far away. I've always been confused about stuff coming from below the bud union. Most of my strongest new canes have come from below what I thought was the bud union and they are pure McCartney rose! Is it possible my rose was not grafted? This new thing is very tiny and delicate, it doesn't look like the shoots from my main plant (though the foliage seems the same). Thank you for the input!

I had 2 seedlings to come up this past spring from Playgirl. They were tiny little things when I found them.
Patsy

I took for the first time this year three suckers or unions whatever from main bushes just because and had some roots so put them into large pots with cactus dirt and Supersoil potting mix and they grew nicely. Bet my neighbor that they would bloom and that flower was small, but worth the experience but that Dr. Huey is right on because I have a huge eight foot plant that was here when we moved in and it's first bloom is a knockout and that's the whole story for the whole season except it doesn't like poison or fertilizer, but I can't that huge thing out where it is so I just leave it. But as the cuttings are growing and if the leaves stay small and the blooms are small and unimpressive I won't keep them. You take a shot and might start a winner that you will be proud of as your own. I have quite a few cuttings that have worked and become real show stoppers.

Well I finally decided to start digging down and I'm immediately discouraged. The little plant is almost three feet away from my McCartney rose, could it really be a sucker? I feel like I'll have to dig forever to try to trace it back and I'm worried about damaging it if it is not in fact a sucker. I think perhaps I'll wait to see if it blooms this year.

I read about your rose seeding and was wondering what it turned out to be. Also I am been looking for carpet roses and can not find any here locally. Would you be will to trade some of yours for some of my yellow lady bank thornless roses? Let me know. Thanks in advance and may GOD bless you and yours, your garden and the USA while we still have a Country left.