Refurbished MacBook Pro 2014 turns off without keyboard or trackpad interaction

I think it sounds like you bought a damaged and malfunctioning laptop. "Refurbished" in fact indicates that something was known to be broken at some point in the past. It sounds like something is quite wrong with the power button -- it doesn't work properly when starting up the Mac and doesn't work right after the Mac is running, it's as if it is being pressed all the time. And other elements of the keyboard seem to be not operating properly. I would guess that someone tried to do some work on or under the keyboard and damaged it in the process, which is easy to do as there are many tiny circuits and wires.

The laptop is 11 years old, which is way past its expected lifetime. Apple won't work on it (obsolete) and doesn't stock parts for it. In fact parts for it are probably impossible to find. I would guess to repair it would cost as much as a new MacBook Air, which starts at $999 ($899 in the education store). Apple and OWC also sell properly reconditioned Macs at good prices and they come with warranties, so you aren't stuck with someone else's problems.

If at all possible I would return this Mac to the seller for a refund as you received a damaged and malfunctioning product. Diagnosing and repairing it is not practical to undertake.

2014 is long in the tooth.

see vintage and obsolete:

Obtaining service for your Apple product after an expired warranty - Apple Support

Obtaining service for your Apple product after an expired ...

ref: Certified Refurbished Products - Apple

Refurbished here can be open box returns, not what a buyer wanted, or wrong order as well as issue that have been repaired/replaced. They all carry the exact same 1-yr limited warranty, with the 14 day return poilicy. With the same option to buy the AppleCareOne (AppleCare+) extended warranty.

14 days for an exchange or refund for any reason—see Returns & Refunds Order status

Returns & Refunds - Shopping Help - Apple

Only items that have been purchased directly from Apple, either online or at an Apple Retail Store