Wordology

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.