If I control a Lion's Eye Diamond, and I have a Paradox Engine in my hand I wish to cast, can I put it on the stack & crack LED for mana to pay for Engine; or do I have to crack LED first, losing Engine before I can cast it using the mana from LED?
You can't activate Lion's Eye Diamond's ability while you're casting a spell. You can activate that ability "only as an instant" (see the Oracle text [C.R. 108.1]; this restriction is not given on the card as printed in Mirage) (C.R. 602.1b), which means that you must have priority to activate that ability (C.R. 304.5). Notably, no player has priority in the middle of casting another spell such as Paradox Engine (including at the moment you can activate mana abilities under C.R. 601.2g; note that C.R. 601.2a-h don't provide a priority window).
EDIT (Jun. 19): Added rule citation.
EDIT (Apr. 20, 2021): Edited, including to conform to rule changes with Strixhaven.
EDIT (Jul. 2, 2021): Edited generally.
Normally, you can activate mana abilities in the middle of casting a spell in order to pay for that spell. However, Lion's Eye Diamond has this specific clause in its Oracle text: "Activate this ability only any time you could cast an instant.". Since you can't cast an instant while a spell is being cast since no one has priority, this clause prevents you from cracking Lion's Eye Diamond to pay for a spell your are in the middle of casting. You need to crack it first which means you will discard your hand before being able to put the spell on the stack.
Also, I have two floating mana to pay for Engine.
EDIT (Jun. 19): Added rule citation.
EDIT (Apr. 20, 2021): Edited, including to conform to rule changes with Strixhaven.
EDIT (Jul. 2, 2021): Edited generally.