Does SOAP API support API request or response similar to how REST does using compression headers?
Yes, it does.
If a client compress the request and includes a
Content-Encoding HTTP header with the name of the compression algorithm (either gzip or deflate), Salesforce decompresses it before processing. Moreoever if the client includes a
Accept-Encoding HTTP header (either gzip or deflate), Salesforce should compresses the response.
It's stated in the Implementation Considerations page. Here are some relevant part:
Compression is not used unless the client specifically indicates that it supports compression. [...]
To indicate that the client supports compression, you should include the HTTP header “Accept-Encoding: gzip, deflate” or a similar heading. The API compresses the response if the client properly specifies this header. [...]
The API supports deflate and gzip compression according to the HTTP 1.1 specification. [...]
The API is not required to compress the response even if you have specified Accept-Encoding, but it normally does. If the API compresses the response, it also specifies a Content-Encoding header with the name of the compression algorithm used, either gzip or deflate.