For #1, see my question here about that issue generally; the consensus is that it will make combat less of a choice since all archers are also melee combatants with no consequence. I'd suggest maybe not doing this, but it's not a huge problem if you do it anyway; maybe see how much of a problem it is and implement this if not doing this is impacting your fun.
For #2, this should already be the case anyway, since this essentially counts as an important NPC, and PHB pg. 198, under "Monsters and Death", says:
Mighty villains and special nonplayer characters are common exceptions; the DM might have them fall unconscious and follow the same rules as player characters.
For #3, not issuing a command; the spell Conjure Animals (PHB, pg. 225) says:
They obey any verbal commands that you issue to them (no action required by you). If you don't issue any commands to them, they defend themselves from hostile creatures, but otherwise take no actions.
which sets a precedent for beasts defending themselves (which could easily be extended to protecting the ranger so long as they only did so whilst remaining by the ranger's side), so I'd say what you propose in your question seems fine to me. (Thanks to @V2Blast for reminding me of this quote)
Finally #4; there are a few dnd-5e questions about changing how this works; the most relevant of which is likely this one (again, thanks to @V2Blast).
I can say that when DMing for my partner (whose first character was also a Beastmaster ranger), I suggested that it uses up her character's Bonus Action rather than Action, which means it still has a cost, but not one has harsh as using up an Action. Commanding the beast as a free action removes any cost, so I'd recommend a Bonus Action, which worked well for us and felt balanced (i.e. want to cast Hunter's Mark? Your beast does nothing this turn).
I know that the quote above from the Conjure Animals spell says "no action required by you", but in this case you've already paid the price of a spell slot, and it's a 3rd level spell, which for rangers specifically requires them to be level 9, so it's a big difference from an "always-on" class feature that you have from the level 3 Beastmaster archetype. Hence I'm sticking with my advice of having commands cost a Bonus Action.