That lower growth is coming from below the graft. Most citrus are grafted onto the rootstock of the Trifoliate orange (Poncirus trifoliata). You'll need to keep those suckers pruned off as soon as you see them, or they will very quickly take over. It is not unusual for this to occur, and will probably stop after some time. Meanwhile, just keep after them with sharp pruners. That growth will sap the energy from your desirable lemon cultivar.

A chute is just a new TP limb (or scion) as it emerges from the ground. If you look to the left of the large limb of the TP, the smaller limb is a new chute that has emerged since I planted the TP in October. Just to the front of the smaller limb and just behind the larger limb you can see a reddish tint in the mulch. Those are also new chutes emerging. This TP arrived with 3 large scions. My labrador stepped on it chasing a squirrel and broke 2 of the 3 scions off. I have taken the two scions and mated them with two herbaceous roots. We will see what happens with them in the spring. But, the new chutes are scions as they emerge from the TP roots. Hope this helps.

One thing I found out the hard way about is that moving them causes many of them to die off, and even the ones that survive are droopy for a while, obviously being stressed so the lesson I learned was that even if the seedlings come up very close together, they should never be transplanted until they are dormant in the fall, perhaps by November. I have also learned from another forum member that azomite is supposed to be good for them. Other than that, it seems like the best I can do is keep the weeds out, and water them if it gets too dry as it seems like they will only produce 1 or 2 leaves on their first year. It kind of sucks that the ones in this picture will be fully done growing for the year by the middle of April. Here is a link that might be useful:

Most tree peonies are grafts. What you are seeing is the nurse root sending up foliage. They will remove food from your tree peony. If they have buds I would allow one or more to bloom to see what the flower looks like. Couple of questions to ask yourself. Is the herbaceous flower something you want to keep? Sometimes they are quite nice. Did you plant the tp deep enough that it has developed it's own roots? You have a couple of options. Cut or pull the herbaceous stems now but you need to dig down carefully to see if you have roots coming off of the upper stem. This fall you should dig the plant checking for the tp's own roots on the upper part of the stem. If the roots are there cut the nurse root from the tp. Keeping if you like, give to someone, compost. Replant tp a little deeper so it will develop stronger roots. If there are no roots from the tp you need replant deeply with much of the tree peony stem below ground. Do not remove nurse root because the only nutritian the plant is receiving is from the nurse root. The idea is to plant the grafted tree peony deep enough that the nurse root dies and the tp portion sends out its own roots.