1) yes that's fine. You can also say "I'm on the road with a Walmart", or "I'm on the road next to Walmart". They all work in English.
2) That sentence makes perfect sense.
3) You would say "I bought a Ralph Lauren shirt". Although Ralph Lauren is technically a person, you are referring to the company, not the man himself, so you don't need the -'s.
I would say "That shirt is Ralph Lauren" or possibly "That shirt is from Ralph Lauren"
Again, if I was referring to Ivanka Trump's brand, rather than her personal dress, I would say "I bought
an Ivanka Trump dress". I don't think you need the word 'brand' in this context, as it seems obvious this is what you mean, but you could say "I bought an Ivanka Trump brand dress". Either way, I wouldn't use the -'s at all.