A child like your stepdaughter will not respond positively to spanking. Try love and attention when she is being good, ignoring and putting her in her room when she is bad. Don't argue with her or yell at her.
Two year olds often cling to their mothers. That is normal. If he needs a hug and kiss first, then I think she should give him a hug and kiss first. That won't hurt anybody. If children have a meltdown in a store, the best thing to do is leave right away, and go home and give them a time out in their room to calm down.
The two of you have two children together. I assume you are quite young. Are you going to get married and make this a permanent family? Your kids will respond positively to that. Kids need stability, and they thrive on knowing their lives will be stable.
Raising children is REALLY hard. You have to have a lot of patience. Don't assume that because children try your patience, that there is something wrong with them. Most of the time, changing how you parent them is what does the trick. Parenting classes are helpful. Good luck.