The API and ASME standards have in common that they cover pipeline equipment (and valves). However they have a different origin.
The API specifications are focussed on upstream equipment, like wellhead equipment, X-mas trees and subsea piping.
The ASME specifications are focussed on downstream equipment, like separator valves, transport lines and processing plants.
It is possible to design a valve that complies with both specifications, but it is very unlikely that an API valve will always cover ASME B16.34 standards.
Example:
API 6A calculates strength based on material classes (45K / 60K etc.) Wallthicknesses may vary, depending on the material.
ASME B16.34 tells you the minimum wallthickness and then provides tables what the strenght will be at what temperature.
So my advise is to keep the standards separate. Do not purchase an API valve if you need an ASME valve or the other way around. If you are the manufacturer: go through the standards in detail and tabulate the requirements for each pressure class. There is no easy way out.
In general the valve design is not restricted by API. These can be all kind of globe or gate valves. But note that manufacturing valves in accordance with API needs API approval/verification. This may restrict the possibilities.
Example:
At high pressure classes API may require as a minimum 60K materials. If a typical design can not be manufactured in high strength material, you will not find it as API qualified.

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