I recently purchased this game & after a struggle with the manual and 2 actual playthroughs, we had encountered a situation for which we couldn't find a solution.

This is the situation:

I had played the n°2 order card (activate hex, move troops in from up till 2 spaces away - supremacy: do the same, but no battle allowed)

Step 1 was a success & I had supremacy.

The enemy had retreated from previous hex & had 2 bowman standing up in the hex where he had retreated to. With my hero and his object, I was able to rout the remaining 2 bowman, and then I rushed in a berserker of mine.

Question is, is that invasion still considered as a battle or not?

Since routed units in a hex can not partake in a battle and there were no more other units present, I was inclined to say it was a legal move...

Any of the more expirienced players who can shine a light upon this?

According to the rules - as soon as you meet up with the other player on the same hex battle happens, so I would say yes, it is still considered as a battle.

Additionaly - routed units would still participate in battle, they just would not be able to do any damage and count its straingh in the end, so you would have better chance to finish them up.

It is definitely still considered a battle. Anytime you move into an enemy-controlled area, it's a battle. This is even true if you move into an enemy area that doesn't have any units (like an empty stronghold or an empty home area).

Technically speaking, you'd still draw cards for all your units, too (although in most cases, it doesn't do much. However, units like the Knights and Necromancers can make use of those cards). It does count against the 1-battle-per-Mobilize even if there are no enemy units there.

One note, however - if you move into an area that contains previously routed units, those units are NOT part of any battle that may happen there. If the defender loses the battle, those units will be destroyed, but unless you stand those units up before the battle starts, they can't participate in any way. (Units that are routed DURING the battle are still part of the battle, in case you have a card like Rally Cry; but Rally Cry won't let units that were previously routed join the battle after it's already started. You could play the card right at the start of battle and they would join in, but if you wait beyond that, it'll be too late).

So Mobilize cannot be used to occupy two enemy home provinces that are completely empty?

Correct. That would constitute two "battles".

(Thematically, it could just be assumed that the home land areas have some paltry defenses, not enough to do much damage, but enough to slow your armies marches).

There is a bit of a gray area here as RAW (rules as written) units that were routed BEFORE combat do not participate in the battle, they stay on the board and don't join the board. So is there a battle if there are no units participating on the enemy side? Same happens with Strongholds as the definition of Battle is between opposing UNITS (as per the rule book at the very beginning of the battle section) when one side has no units is it considered a battle even if tally strength is still needed? Only reason this really comes up as mentioned it the Daqan and the Undead can benefit from it. Since the undead are already the lords of combat vs anybody except elves we are inclined to interpret it otherwise, but we are looking to see how other people handle this.

Yes, but a battle takes place if you enter into a territory that is not friendly or unoccupied (unless attempting diplomacy to neutrals). Since there are enemy units there, the area is controlled by an enemy, even if all the enemy units are routed. That means a battle still takes place and you still draw all the fate cards for your army.

In the example given, the defending player might use a Summon Lightning card, killing off the berserker and winning the battle (the bowmen were not in the battle so would not get damage and the resulting 0 - 0 strength tally would be in favor of the defender).