Is there a way to have your bombardment units indiscriminately fire on any and all enemy formations or do all bombardment units in support/rear echelon have to be assigned to support a specific formation in order to attack?
If your bombardment formations are not assigned to support, they will do counter-battery fire, basically they will fire at any enemy bombardment element within range. However there is not an indiscriminate fire option unless you want to put your artillery in the front line (please do not put your artillery in the front line).
Are light bombardment elements that fight on the front line less likely to be targeted by enemies?
No.
Do they still fire on attacking enemies as if they were supporting?
Light bombardment elements in a front line attack/defense formation fire as normal units. They are small and cheap enough that this is an okay strategy if you want to use them to represent infantry mortars, however they can also work from the support echelon. Note that LB are not generally capable of counter-battery due to short range so if you use them in the support echelon they need to have a support relationship set.
If a bombardment unit is set as "non-combat," does it suffer the malus to its chance to hit when bombarding or only when it's on the frontline?
It will suffer the malus. The only units which should be set as non-combat are those which do not fire in ground combat (note that this includes STOs, since they do not fire in ground combat, even though they are a type of weapon).
Do units with static headquarters, or really any static units, completely miss out on the ability to do breakthroughs? For example, for a tank formation with a static HQ, can the tanks still do breakthroughs?
Breakthroughs occur on a formation basis, so a formation with static elements can break through but will have a lower chance to do so due to the static elements.
Can static elements attack or only defend?
Any formation can be set to any stance, including front line attack and defense. Static formations are no exception. The difference between front line attack and defense is only that attack uses unit evasion/hit modifier stat while defense uses unit fortification stat, and that attack stance formations have a higher breakthrough chance (factor of 2x).