1.) Should a warlord be able to restore hit points in combat?
Yes, unequivocally. Otherwise it is a strictly inferior support class, and non-viable. Insisting the Warlord not be able to restore hps in combat is identical to insisting no one ever be allowed to play one.
Sorry if that sounds strident or uncompromising, but suggesting the warlord be strictly inferior is very strident and uncompromising, and invites such a response. This should not even be a question. Asking it creates the appearance of bad faith. Besides, it's been asked and answered many times.
2.) Should a warlord be able to restore hit points using only his voice?
That's ambiguous. The Warlord inspires. That /could/ be a matter of verbal communication, or it could be a wordless shout, it could be a shinning example that allies need only /see/, it could even be an ally remembering something the warlord had previous said or done in a new light.
3.) Should a warlord be able to restore hit points using only his voice when the target is unconscious or "wounded"?
Yes, unequivocally. Otherwise it is a strictly inferior support class, and non-viable. Insisting the Warlord not be able to bring a downed ally back into the fight is identical to insisting no one ever be allowed to play one.
Sorry if that sounds strident or uncompromising, but suggesting the warlord be strictly inferior is very strident and uncompromising, and invites such a response. This should not even be a question. Asking it creates the appearance of bad faith. Besides, it's been asked and answered many times.
4.) Should a warlords ability to restore hit points be considered magical or non-magical?
It should be non-magical per the rules. It could, however, be 'considered' by individuals both in or out of character, to be otherwise. You can always write-off conclusive evidence if you have sufficient belief. An atheist PC could be convinced that Clerical spells are 'no different' from Wizard spells, and do not 'prove' the existence of deities. The Cleric class need not be undercut in any way to allow that.
There is a belief this doesn't gel with 5e's interpretation on hp and healing though
That 'belief' is proveably false. First of all, it's absurd on the face of it. 5e has no interpretation of hps and healing, it has mechanics. So it's nonsense. 5e has one side-bar that mentions one possible way of describing hp damage that is in no way incompatible with what the Warlord has always done. That's it.
Secondly, those mechanics do not imply anything that would prevent the Warlord from being a functional addition to them. Non-magical healing can already be up-from-0 (Death Saves, rest & time, healing) and can already be done in combat (Second Wind, healer feat). Hit points can be interpreted as almost wholly or only partially non-physical, but they cannot be interpreted as modeling serious all-meat injuries that must be totally healed or magically caused to vanish to restore a character to full hps - HD, overnight healing, second wind, and death saves all contradict such models.
but people disagree on which limit they wish to place. For some, they'd be fine with 1-3 if 4 is answered "magical". Others say if 1-2 is fine for non-magical, but 3 is magical. Others things 1 should be nonmagical, but 2-3 magical. Some thing the warlord shouldn't be magical at all, while others are fine with a little magic if it means getting more options. In the end, someone is going to lose.
No one has to lose. The Warlord could be introduced in a reasonable form, true to it's concept, and effective & balanced. Those who are already OK with hps & healing as-is in 5e would be able to use it unchanged, or with the caveat that some people choose to attribute some stuff to magic even though there's no mechanic to force that conclusion. Those who use models that make some aspects of the warlord impossible would already be choosing modules and banning classes or specific abilities to bring 5e in line with those models, and could do the same with the Warlord, either banning it outright or tweaking either function or explanation at the mechanical level.
Just because some people who would never use the class, or (in spite of constantly arguing against it) 'don't care' about it, have spurious objections to some aspects of the concept doesn't mean it shouldn't be allowed into the Core game. 5e already allows anyone that has a mechanical or conceptual problem with a class to deal with it: by simply not using the offending concept or mechanic, or by changing it as a DM. This would be no less true of the Warlord than it already is of every class.
Question: Is a Barbarian "Magical" or "nonmagical"?
Inclusive or?
Yes.
The Berserker is not stated or implied to be anything magical or supernatural, while the Totem archetype calls upon spirits to perform magical effects.
See, Rage to me is magical
Nothing calls it out as magical, though. So it's fine to have a class presented as non-magical, that some folks might be more comfortable conceiving of as somehow magical, even though it doesn't cast spells or lose abilities inside an anti-magic shell. The Warlord doesn't need anything more than that. It doesn't need to be in-your-face, unequivocally, no-alternative-possible, non-supernatural-in-any-way (and, in 5e, /couldn't/ because of DM empowerment).
You need have no more issue with the Warlord than the Barbarian, in that case.
Warlord should be the same way. They tap into something beyond mortal abilities. They don't cast spells, they don't learn magic, but they are magical in that they can do things far beyond what a normal soldier can do.
Being able to do more than an ordinary person does not require magic. You're free to imagine it does, and attribute magical powers to people who exceed the norm - like Olympic Athletes, victorious warriors, rock stars, or whatever. It's not an unusual human compulsion, even IRL, where magic (we presume) doesn't exist, at all. In the context of a typical D&D world, attributing supernatural power to high level characters of the very few sub-classes that don't actually have them would be quite natural.
