Glad to see someone interested in the field. Ok one question at a time. Civil Engineers do lots of things. Generally we focus in 5 big areas: structures, construction, geotechnical, transportation, and water resources. Sometimes surveying is involved too. Usually, undergraduate work is more general and you take classes to apply to each of those areas. If you want to specialize most people go for graduate study in one particular area.
Structural engineers are all about design and strength of bridges, frames, trusses, building systems, etc. Construction engineers focus on construction methodology, scheduling, optimization, and management. Geotechnical engineers focus on foundations, settlement, strength and properties of soils, pavements, and subsurface features. Transportation engineers design roadways, traffic control systems, safety systems, mass transit, shipping and freight facilities, etc. Water resources engineers do drainage design such as ditches and culverts, detention ponds, lakes, etc. They also design drinking water and wastewater collection, treatment, and distribution systems. These are only a few of the things Civil Engineers are involved in and most civil engineers cross over between areas to some extent. The great thing about the civil engineering is that your background is usually general enough that you get to work to some extent in all the areas as your career progresses.
As far as the design work its a grab bag. Some engineers do just design, but some do only field work, data collection, construction site management, and inspection. It all depends on your interest and the project you're working on. Ideally you'll get to do a little of each to keep it interesting. How much of the design do you know? Well you know parts or have an idea of what to do, but it isn't necessarily like you pull something off the shelf and use it over and over again. There's always something new or unexpected. There's also always new materials and building practices to consider. As for your PE, that comes later. First expect to spend 5 years in school (hardly anyone can do it in 4) then take a very difficult intern test. After you're an intern for 4 years on the job you take another test for your PE. More than anything a PE gives you the responsibility to certify that your design is safe and protects human life and all that. Actual rules vary from state to state, but some states let you get a year of credit for completing grad school.
All and all its a very rewarding career. The work can be difficult, the hours are long, the pay is decent, and there's always something new around the corner. If you still want more info, PM me. Hope it helps!