Your meditation method cannot to be too comfortable, nor too uncomfortable, that is why people mostly choose to sit in half or full lotus with straight back so they don't slouch and sloth. Be careful not to slouch, keep back upright, work on posture and correct it.
Question 1:
Attempting to sustain alertness in meditation, not meditating when too
lethargic. In other words, always exerting oneself and keeping
sharpness when in meditation, and stopping when the mind becomes
unclear.
Sustain, but gentle mindfulness or awareness, I wouldn't say alertness as it implies stress as in fight or flight type of thing. Note that you are feeling lethargic. Do not stop, note that mind becomes unclear. Eventually correct your posture, straighten the back and continue. It is generally to be mindful of it just to the extent of eating, or waking up (both in meditation and daily life)
Again, bhikkhus, a bhikkhu is one who acts in full awareness when
going forward and returning; who acts in full awareness when looking
ahead and looking away; who acts in full awareness when flexing and
extending his limbs; who acts in full awareness when wearing his robes
and carrying his outer robe and bowl; who acts in full awareness when
eating, drinking, consuming food, and tasting; who acts in full
awareness when defecating and urinating; who acts in full awareness
when walking, standing, sitting, falling asleep, waking up, talking,
and keeping silent. 9. “In this way he abides contemplating the body
as a body internally, externally, and both internally and externally …
And he abides independent, not clinging to anything in the world. That
too is how a bhikkhu abides contemplating the body as a body
Question 2:
Or, engaging with meditation even when lethargic, continuing one's
meditation despite the mind being unclear and hazy. In other words,
waiting for the mind to clear up.
Engage, yet gently noting haziness and confusion, lack of clarity. Eventually correct your posture, straighten the back and continue.
On mental hindrances, such as:
- Sensual Desire
- Ill-will
- Sloth and Torpor
- Restlessness and Remorse
- Doubt
https://www.accesstoinsight.org/lib/authors/nyanaponika/wheel026.html
Bonus: "Is there such a thing as bad meditation?":