So I am an independent contractor with and LLC in the US. I periodically reside in Spain and want to offer consulting services to Spanish and other European clients that would be conducted online/via Skype. What are my tax obligations in US? Spain? Europe? Thank you!
With something this complicated you are going to want to consult professionals. Either a professional with international experience, who will tell you the best tax arrangement overall but might come expensive, or one professional in each country who will optimize for that country. You will have to pay US taxes, and depending on your residency probably some in Spain. Double tax agreements should kick in to prevent you paying tax on the same money twice.
You do not have to pay separate 'European' taxes. If you do substantial business in another country you might have to pay there, but one of your professionals should sort it out.