Push the envelope belongs to the modern era of the
airplane. The “flight envelope” is a term from aeronautics
meaning the boundary or limit of performance of a flight object.
The envelope can be described in terms of mathematical curves
based on things like speed, thrust, and atmosphere. You push it
as far as you can in order to discover what the limits are.
Go haywire relates to actual haywire. In addition to
tying up bundles, haywire was used to fix and hold many things
together in a makeshift way, so a patched-up place came to be
referred to as 'a hay-wire outfit'. It then became a term for
any kind of malfunctioning thing. The fact that the wire itself
became easily tangled when unspooled contributed to the use of
the phrase.