What do you have to offer in exchange?

Definitely a bad graft with the standard cherry suckering. This happens to a lot of ornamental fruit trees. You'll have to cut the white flowered trunk off at the collar (just past where it meets the main trunk). It sounds like both trunks are the same size? That means this problem started as soon as you planted it. A picture would help a lot. Make a CLEAN cut on an ANGLE just past the branch collar. It is probably too thick to use loppers, so purchase a decent pruning saw (Lowes, HD sell these), and make a starter cut at the bottom of the branch, angling up and in toward the tree, then do the rest of the cut from the top of the branch, angling toward the bottom cut (away from the tree). The cut is on an angle to shed water. You must do a cut on the bottom first so that when you cut from the top it doesn't pull down and rip the bark along the main trunk, which can kill the tree. Do this now or wait until next year and do it in early spring before the buds break. Do not wait much longer, because right now the cherry will use it's energy to heal the wound, but if you wait until May or later it will have used most of it's energy in flowering and growing new branches and will not heal quickly at all. The standard cherry will grow about 20 feet high. Depending on the rootstock, it may even bear fruit. Standard cherries are hardy trees, and it will definitely end up killing your weeping graft if you don't get rid of it soon. As for it looking odd, don't worry, in a few years the weeping tree will grow to the other side with proper pruning. Right now the other tree is preventing it from growing that direction. Once it is gone the other side will start to thrive. The sun exposure alone will help a lot. Don't expect too much this year though, since it will be focusing its energy on healing.

I'm not an expert, so you will have to google it unless someone else chimes in. But those are aerial burr knots which m111 are prone to grow. My trees are 10 years old and they have outgrown them, I.e., it has not been an issue. No pun intended. I did have to battle knot borers but the latex paint wash at the base helped. It is less of an issue now as the trunks are bigger

Give yourself some time to read up on proper pruning technique- it's easy to ruin a tree if you make poor choices. Aesthetic pruning is an art as much as a science so take your time and study up. Many nurseries used to offer pruning classes pre-covid, I'd look for something like that or a good book to help you understand why you are cutting where and how each cut affects future growth. When the day does come for pruning (I like very early spring for this so you can see what you are doing) take your time. Stand back, have a vision, and don't rush. I spend way more time standing and staring making my decisions than I do cutting for sure, lol. But you can always come back the next day and take more off but once you make a mistake it can take years to grow out.