Not exactly, although at first the Founders did try to build an EU-style federation.

[The Articles of Confederation and Perpetual Union] (http://avalon.law.yale.edu/18th_century/artconf.asp), normally just called the "The Articles of Confederation" were much closer to an EU-style system in that sovereign states provide mutual benefit to each other and citizens of one state are "entitled to all privileges and immunities of free citizens in the several States".

Article II says:

Each state retains its sovereignty, freedom, and independence, and every power, jurisdiction, and right, which is not by this Confederation expressly delegated to the United States, in Congress assembled.

The Articles delegated few powers to Congress, it even lacked the power to tax.

States even specifically retained certain powers. Article VI says that with Congress's approval states could make treaties with a foreign "King, Prince or State". Under the same article and also with the consent of Congress states could "send any embassy to, or receive any embassy from" a "foreign "King, Prince or State".

Congress also lacked the power to control interstate commerce and foreign trade, so each of the thirteen states had their own trade policy with foreign governments and with each other.

Obviously the articles were not perpetual and The Constitution was intended to bind the states into a "more perfect union." In creating that union some very specific details were included in the Constitution to fix flaws in the Articles.

A lot of what you ask, including every state having two Senators, were part of a compromise to create a national government agreeable to all the states. This ["Great Compromise"] (https://en.m.wikipedia.org/wiki/Connecticut_Compromise) balanced the needs and wants of small and large states. Specifically to your question, the bicameral legislature where representation in one chamber is based on population, the House of Representatives, and a second chamber where all states are represented equally, the Senate. In both chambers every Representative and Senator has one vote, this specific detail solved another issue in the Articles which was that every state had a single vote. There is one exception to every Representative and Senator having one vote. When no candidate wins a majority of the Electoral College, the House of Representatives elects the President and each state gets a one vote.

The Electoral College has a complicated history. That history has less to do with the states being sovereign and bound in an EU-style pact than it has to do with the framers opinions of "the people," political compromise and horse trading, small versus large states and most of all slavery.

Since the EC's history does not go to the heart of the question I will not address it any further here.

I said that the Constitution contained specific text to fix flaws in the Articles, interstate commerce is one of those flaws that needed correcting.

Under the Articles interstate commerce was at best problematic. Not only did foreign countries have different trade relationships with each of the thirteen states, because each state could separately enter into a treaty with a foreign country, states had different trading relationships with each other. The situation resulted in [states treating each other as foreign countries] (http://neusysinc.com/columnarchive/colm0063.html) including imposing "discriminatory taxes, licensing provisions, port regulations, and the like."

As for the word "state" it remained because Constitution united thirteen existing states.

These governing entities were already sovereign states, what else would they call them and why would they even want to change it? At this time people were loyal to their state first and this new United States of America second, there was no national identity. Plus I imagine that were far bigger problems the delegates faced than coming up with a new term for "state."

So was an EU-style system envisioned, at first something like that was implemented, however it did not take long for that EU- style arrangement to disintegrate. The resulting government under the Constitution includes very specific "fixes" for problems that arose under the Articles and required sovereign states to cede enormous power to a central, national government.