Hi L..
It sounds like your daughter is smart.
At this age they are very literal. If you said you were going to buy a helmet and didn't tell her there would be other stops along the way, she was probably very upset with you. I would say the 'walking' thing was her attempt to glean 'literal' understanding because she was frustrated that you weren't at the helmet store yet.
I have a kid like this and if you can figure out how to manage her it will get better not worse.
For starters..... I am very up front and specific with my daughter about information. It's better now that she's almost 11, but it was pretty rough when she was 4 -6..... so my conversation BEFORE we left the house would have been something akin to this.... (generally I do more asking than telling, so that she can answer the questions rather than me telling her what I expect..... this is just an example)
"we are going to go shopping to get a helmet so you can ride your bike. But before we do that I have to run 3 or 4 errands and you will come with me. While we are at those 4 stores I will expect that you walk right beside me and behave. If you do that I will be able to get my errands done quickly and we can get to the bike store for your helmet. If you don't have acceptable behavior we won't be able to go to the bike store today for a helmet." Then it would be repeating parts of this during the trip
"we've been to 1 store - how many left until we can go to the bike store. yep that's right 3 more."
"Oops - I totally forgot that I need to pick up a picture for Aunt Louise. I need you to put your flexible shoes on.... we have to add a stop. That's 5 total stores and we've been to 2.... so how many more til we go to the bike store? yep that's right 3.
etc etc
As far as handling the walking issue as it happened I would have handled it like this....
Your situation.....
She refused to walk the same pace as me, if i went slow, she went slower, if i asked her to walk next to me she would bolt ahead of me and say "like this? Is this fast enough!?" .......
I would have said......
"no, susie, that is not the appropriate behavior. I asked you to walk next to me.... are you walking next to me? (let her answer and correct the behavior. if not then you say ) Nope, you are in front of me. If you can't walk next to me while we are in mommy's stores we won't be able to go to the bike store for your helmet. Please walk next to me now".
Good Luck!
B.