Firstly, it is worth noting that (as Eidolon notes) temporary hitpoints do not stack (pg.198 PHB). That is to say, if you have 10 temporary hp, and gain 12, you now have 12, NOT 22.
Technically yes
With that in mind, as per this question, all actions that can be used in combat, can be used outside of combat. Combat is not special in that regard - it is just a way of resolving encounters in which the participants want to fight each other. So, RAW, maneuvers can be used outside of combat.
But a DM could reasonably disallow it
However, it could be argued by your DM that 'bolstering companions resolve' outside of combat is inneffectual as there is nothing around to damage their resolve. Hit points, including temporary hit points, represent not only physical strength and wounds, but also mental resolve. Gaining temporary hit points through rally represents having your mental resolve strengthened by an inspiring shout from the fighter. A DM might argue that, since 'rally' implies rallying characters from fear, this would only work in combat.
That said, I would be tempted (as a DM myself) to allow it on the basis that an inspirational speech before battle could bolster one's resolve. There is a reason why commanders of (e.g. ancient) armies gave speeches to their men before battle. Given that, as noted above, temporary hit points do not stack, this should not be too overpowered.
No. You cannot forgo attacks you can't take
You can take only 1 attack
Say your character can normally make 2 attacks by using the extra attack feature:
Beginning at 5th level, you can attack twice, instead of once, whenever you take the Attack action on your turn.
The net's description says:
When you use an action, bonus action, or reaction to attack with a net, you can make only one attack regardless of the number of attacks you can normally make.
When you attack with a net you now can take only one attack regardless of your extra attack feature.
Thus you "have" only one attack now. It is worth noting that "number of attacks you have" is not something the game ever says. The only thing that is defined is the number of attacks you can take. And that number is now 1.
You are willingly foregoing those extra attacks as a cost of using the net.
You cannot forgo attacks you can't take
When you take the Attack action on your turn, you can forgo one of your attacks and use a bonus action to direct one of your companions to strike.
Notice the word "forgo" which means "omit or decline to take". You cannot "decline to take" something that you could not have taken anyways. Like you cannot decline an invitation that was not given to you in the first place.
Since you've already used the net, you have no available attacks left to you. Thus, you have no attacks you can forgo.
Seeing as the Commander's Strike feature needs you to give up an attack and you have no more attacks to give up after using the net, then you cannot use commander's strike after using a net.
A heavy crossbow would work
Let me try to make this a bit clearer or more convincing using an example that does work.
Heavy crossbows have the loading property which says:
Because of the time required to load this weapon, you can fire only one piece of ammunition from it when you use an action, bonus action, or reaction to fire it, regardless of the number of attacks you can normally make.
This is very similar to the wording of nets with the major difference being that the loading property says "you can fire only one piece of ammunition". This means that you could easily drop the weapon and continue attacking with another one given the appropriate number of additional attacks. And thus one could also use commander's strike because you still have not spent those potential attacks.
But a net will not
However, the net does not make this statement instead just saying "you can make only one Attack". The easiest way to read this in light of the above is that the net takes so long and is so unwieldy to use that you have to use your entire attack action to wield it.
tl;dr
By using a net you are spending any additional attacks you have to attack with a net. You've essentially spent them and you cannot spend then again. The idea of Commander's Strike is that is that you give extra attacks that you could have made to someone else. But in this case you could not even make those attacks and thus do not have those attacks to give.
Best Answer
They are special nonmagical techniques based on being a scholar of combat. This is explained in the Battle Master entry:
Individual maneuvers explain what you're actually doing:
In each case, you're mundanely directing your ally or goading (through voice or gesture or just annoying attack placement) your enemy. You're just really good at it because of your battle mastery.
Although I can't find a reference in the PHB, it's generally accepted in D&D that you're doing more with your turn than your single attack. Your PC might constantly be feinting, dodging, making threatening jabs, and so on. Your attack rolls represent those actions that are actual opportunities to do damage. From this perspective, then, a Commander's Strike doesn't mysteriously make your ally move faster; it just gives them an extra opportunity that they otherwise might not have had, just like a retreating enemy lets them make an opportunity attack.
In the games I've played, these maneuvers have been explained however seemed appropriate in the situation and were never assigned advantage or disadvantage, but I don't see a problem with the saves being modified in unusual circumstances. I'd just make sure that you're treating other players' unusual abilities with the same level of scrutiny.
P.S.: The Battle Master is, in part, 5e's version of the Warlord class from fourth edition. That class has the Martial power source, indicating that it's wholly nonmagical, with the following explanation: