Was there already a network of freeways that were repurposed/updated/renumbered to be incorporated as interstates, or did the entire system need to be built from scratch?
The interstates were not the first superhighways; by the time a nationwide system was under construction, superhighways in many urban areas had been in use for 30 years. Even in rural areas, several divided, grade-separated sections had been built as interim improvements on many cross-country routes. When the 1956 highway act finally funded construction of the Interstate System (which Congress had technically created a decade earlier), about 1.5 percent of "federal-aid" and toll highway mileage was already to superhighway standard; today—though the statistics are not strictly comparable—about 19 percent is.
The concept of a limited-access, grade-separated, multilane facility was demonstrated in the 1930s with short urban stretches in and near New York City and Chicago's Lake Shore Drive. Several similar facilities were in place around the country by 1940, when the Pennsylvania Turnpike demonstrated the demand for cross-country superhighways as well. World War II slowed building quite a bit, except for projects like the Willow Run Expressway near Detroit that were useful for new defense plants.
After the war, the states and big cities were anxious to get their planned superhighways under way. Cities like New York, Chicago, and LA cobbled together what state and local funding they could assemble to begin construction of projects like Chicago’s Congress Superhighway, or LA's Hollywood and Long Beach Freeways. Looking to the Pennsylvania Turnpike as an example, many states turned to toll financing for corridors where traffic projections were high enough, resulting in the turnpikes of the Eastern Seaboard and industrial Midwest. Nearly all of these got Interstate numbers and became part of the network.
As for what the highway network looked like in 1955, after decades of slow but steady improvement, it was mostly (but not entirely) paved, and most U.S. highways were two-lane roads with shoulders. Lane widths were often only nine feet, though, not the 11-14 feet typical today on high-traffic two-lane highways. Some had three-lane sections, with “suicide lanes” in the center for passing. Many geometric aspects—sightlines, guardrails, median protection, intersections—were quite substandard compared to today. Most state highway systems were similar but even less developed; several states had stretches of unpaved roads on their state networks well into the 1960s.
I like Earl Swift’s The Big Roads as the most readable history of the Interstates, but Mark Rose's Interstate: Express Highway Politics 1939-1989 is the more scholarly source on the congressional machinations that led to the 1956 law.