Every headphone has two key specifications: impedance (measured in ohms, Ω) and sensitivity (measured in dB per milliwatt). Together they tell you how hard an amplifier has to work. High impedance means high voltage is needed. Low sensitivity means more power is needed. A 300Ω headphone with 100 dB/mW sensitivity — like the Beyerdynamic DT 880 — needs a real amplifier. A 32Ω headphone at 110 dB/mW — like many Sony consumer models — runs fine from a phone.
Probably not a separate DAC. Modern motherboard audio and smartphone DACs are genuinely good. The component that makes the biggest difference for hard-to-drive headphones is the amplifier — specifically, one with enough voltage swing for your impedance load. A $100 dedicated headphone amp will outperform a $1,000 phone driving 300Ω headphones, not because it is "higher quality" but because it has the power supply to do the job.
Consumer headphones are tuned to a target curve — usually a variation of the Harman curve — that research shows most listeners prefer on first listen. It emphasises bass and softens the upper midrange. Studio headphones are tuned flat, or close to it, so engineers can make accurate mixing decisions. Neither is objectively correct. The question is what you want to hear: music as the manufacturer colours it, or music as it was recorded.
"Endgame" is the audiophile term for the purchase that ends the search. In practice, endgame gear tends to reveal the limitations of your other components, which triggers another round of upgrades. The only genuine endgame is deciding what you actually want from audio and buying the tool that does that job — even if that tool costs $300 and comes in a cardboard box with no unboxing experience.
The Abyss 1266 TC is a remarkable headphone. So is the ZMF Verite Closed. So is the Chord Dave. They are also subject to diminishing returns so steep that the incremental improvement per dollar spent becomes vanishingly small above a certain level. That level is lower than most audiophile forums will admit — roughly $500–800 for headphones and similar for source gear gets you 90% of what exists. Everything beyond that is the last 10%, spread across a very long and expensive road.