1 - Company A is an European company that sells digital products and services all over the world, including the US. The company does not pay US income tax because it does not have an US presence.
2 - Company A opens Company B in the US as a subsidiary. Company B only exists in the US as a company that markets Company A products. It doesn't sell products or services to the American public. It's just there to do marketing.
3 - Will Company A now be liable for US taxation just because it has a US presence, even if both companies are different entities and even if Company B only serves Company A?
There's no such thing "US Sales Taxation". You're probably thinking of VAT that you have in Europe, and there's no VAT in the US. There are sales taxes imposed by various states, counties and cities, each has its own criteria, but they're imposed on sales (usually tangibles) which occur within that jurisdiction (or shipped to it), if the seller is under the same jurisdiction (mostly). Hope that answers your question.
Re the income taxes, then if you have a US subsidiary then yes, it may make you liable for US taxes on US-sourced income. Even if you don't have subsidiary, you may be liable for US taxes on US-sourced income. What is "US-sourced income"? See my answer on this question here.
In my experience with several companies, "tax nexus" is really what you should be looking at and then looking what is taxable within tax nexus jurisdiction. States are short on revenues right now and "tax nexus" is becoming so broad, that things like sales people or having a server in a data center within the state can put you in tax nexus and you will have to collect sales tax.
So in case #1 - answer is - you would not need to collect taxes.
Case #2 - tax nexus will be established, but the devil is in the details, so scope of it has to be determined by a professional.
Case #3 - company with the nexus is liable for collecting sales tax and paying other taxes.
This is where you should not be cheaping out and getting advice from a qualified CPA!