You are correct in thinking the Civil War was the last (it was the bloodiest war in US history) major war in which the most well-known types of units were state-based. For the United States Army during World War I, the concept of permanent infantry divisions was sort of a new thing.

Prior to World War I, the main permanent units of the United States Army were smaller state-based brigades and regiments like you noted. The regiment (usually, and should have) had 10 companies of 100 men, but this varied widely. The brigade was made up of multiple regiments. The other type of large, permanent unit was called the Corps. Brigades were assigned to a division, an then the divisions (as low as 2, as many as 12) were assigned to an Army Corps.

The division (designated with a number) was not a permanent structure, and only existed as an intermediary in between the brigade and Corps.

The United States' first serious experiments with the division concept led to the Maneuver Division of 1911.

The National Defense Act of 1916 allowed the president greater authority to federalize the National Guard, and consisted of other provisions that organized both the Regular Army and National Guard using the newly agreed-upon divisional structure. New tables of organization were published in 1917, and the first divisions were organized throughout that year.

In World War I, the typical infantry division of the United States bad a "square" organization, with two infantry brigades of two regiments each. There was one machine gun battalion assigned to each brigade, with a third separate, motorized, one.

The division also had an attached artillery brigade consisting of two light (75 mm gun) artillery regiments of two battalions each, and one heavy (6" (155 mm) howitzer) artillery regiment of three battalions. US-designed artillery (3", 3.8", 4.7", and 6" pieces) was not ready for the demands of war, and so we mostly used various French 75 mm and 155 mm guns.

The remaining engineer regiment, supply units, medical, ordnance, and veterinary personnel made the division a very large organization, over 28,000 men.

This very nice chart shows the armies, corps, and divisions that were a part of the American Expeditionary Force, with their WWI-era patches

http://imgur.com/9bqDQmD

During World War I, most divisions had men from all over the country; for the most part, the division began to become a non-state-based organization. In World War I, like in World War II, men were drawn from all over in order to fill the ranks of the Regular Army's new "divisions." The National Army (called the Army of the United States in WWII) and its divisions consisted of those men who were conscripted to fight in the war. These divisions in theory had general recruiting areas like the National Guard divisions, but this broke down as men were sent where they were needed most.

The new National Guard divisions were simply amalgamations and re-designations of the state militia (militia was a title used until 1903 and the passage of the Dick Act, named for congressman Charles Dick, when after the passage of the act, "National Guard" was used) units that had existed before, and consisted of men from the same two, three, or multiple-state area, and as a result often from the same town or city. The 1916 amendment (National Defense Act) to the Dick Act also resolves the issue of the Guard serving overseas; they would simply be drafted into federal service.

Sources:

US World War I division organization

http://www.militaryresearch.org/army_wwi_table_notes.htm

Civil War High Commands, by John H. and David J. Eicher

Excellent question!

The shape and organization of the US Army and National Guard were largely in flux flowing the Spanish-American War and the entry into World War I. Since you seem to be largely concerned with state based units, I'll confine my discussion to the National Guard and later National Army.

In the Spanish American War the Army rapidly expanded as it had in the past by relying on volunteers from the states, called US Volunteers. The War Department would establish a quota to for each state to raise a certain number of troops, and they states would typically ask the state's militia units for volunteers. These units were closely related to the militia, indeed in some cases entire companies or regiments would volunteer en masse, but a legel distinct entity. This is the reason why you will find examples referring to the Pennsylvania Militia as an independent entity during accounts of the Gettysburg campaign 30 years earlier. Units from this scheme would be designated in numerical order based on the state that raised them, eg 20th Maine, 69th New York, or 2nd New Jersey Cavalry.

The system worked well for the Civil War, but broke down during the Spanish American War and the extended Philippines Insurgency. Modern war was growing in complexity, and units raised in this manner often did not gain sufficient training (on an individual level USV took in large numbers of men not affiliated with pre-war units) and higher level unit coordination (even if entire units were inducted they rarely had an opportunity to train in maneuvers larger than regimental level).

Efforts to remedy the status quo began under the Roosevelt administration, and were largely pushed through the Militia Act of 1903, better known as the Dick Act. Under the Dick Act, state militia would be organized and trained following guidance issued from the War Department, in turn they would receive substantial federal funding.

The Dick Act was the birth of the modern National Guard, and as such led to large disruption in organization on the local level. Before the act units would be organized in what ever manner the local community and state government saw fit. In some states, notable Pennsylvania, they followed the federal model but that was not the case in most. In one state a company may have 100 men on paper, but in another it may be 200. The Act proscribed an uniform organization, and allowed governors to beginning pooling their units together with neighboring states. A cavalry troop may have shed it's horses and become a signals company. An irregular sized infantry battalion may become a field artillery battery. States tended to pool all their units together in regiments that could be contributed to multistate divisions, but several states were able to organize their own self contained ones (again, notably Pennsylvania and what would one day be the 28th Infantry Division).

In France and Germany, divisions and corps were typically raised regionally, as they had been since Napoleon. They were local enough to develop unit cohesion and esprit de corps, but not enough to risk an Accrington or a Newfoundland as in the British case.

In the British case, regiments raised battalions locally and then the battalions were assigned four to a brigade, three brigades to a division. Until mid 1915, recruitment was voluntary, and then compulsion under the Derby Plan was introduced. Even after that, and after Conscription was introduced in 1916, there were still volunteers, though not anywhere near as many as in 1914.