In the United States, the federal legislature has introduced the Class system, the best implemented by the state of California:
Any e-bike may be assisted by a motor up to 750 W.
...
Now, state and local legislatures have never made a consistent set of laws, so you may ride your e-bike here but be banned from there. (For instance, it is very hard to legally ride e-bike in most of U.S. National Parks, while any e-bike is allowed to ride in any European National Park). Many U.S. residents simply don't care about the whole class system ("Who will ever check me?")
No.
The federal regulation comes from the Consumer Product Safety Commission and is purely applicable to manufacturers who construct products for sale. That law states (in short) that a motorized electric bicycle,
with a 170 lb rider, can not exceed 20 mph on throttle alone, and its power must be
less than 750 watts (not the same as "up to 750w"). This law has no bearing on private individuals or a DIY ebike conversion since a home build is outside the jurisdiction of - and not relevant to - the CPSC. It also says nothing about where an ebike can or cannot go since that is a matter for the individual states to decide. Note the 170 lb wrinkle.
The 3-class system originated in law in the State of California in 2015. It used the EU regulations as a rough framework, and rather than using the L1e-B limit of 4000w, it uses the 'less than 750w' power limit even for the 28 mph Class 3 thanks to the pre-existing federal manufacturing standard. The California law, unlike the federal, does affect both individuals and their behavior which is to be expected given that vehicle codes are administered by states. Last I looked
28 states 36 states had adopted a version of the 3-class system and it is typically a word-for-word (or near enough the variations do not matter) duplicate of the California 3-class system. This has a lot to do with the fact that People For Bikes did the state-by-state lobbying to get this standard put into state law.
In April of 2020, various agencies within the Department of the Interior proposed and later adopted rules for ebikes that included the 3-class system in their definitions of what an ebike is (but not on where they can go). The biggest takeaway from this is the lands are primarily inside National Parks and do not include land managed by the Forest Service. So... only a small fraction of the land available for recreational use in the USA.
Later, in September of 2020, the Department of Agriculture joined in and the takeaway here is they do include land managed by the Forest Service. Rules on whether or not ebikes are allowed, and which classes, are up to local land use officials and subject to the usual public comment periods and appeals. That is all sorting itself out right now, or was sorted recently. Check your local area's rules.