1. Have there been any battle between two faction (Clan and IS) on a planet and suddenly earthquake struck the battlefield?
Any PSR to offset the lose balance of the mech?
There is a scenario in the (apocryphal) German scenario booklet
Bruderkampf where just that happens, plus lava flows. The scenario booklet is available free for download (see Sarna article) and was designed specifically as a mini-campaign for new players using nothing but the starter box set.
Earthquake rules are found in
Tactical Operations, p. 55.
2. How far away are the clan homeworld from the inner sphere. I have seen map that indicate that the clan homeworld are the furthest human inhabitable planets.
Any planets that are further than the clan homeworld? Not counting the ISFT*'s homeplanet.
Nobody in-universe knows what human settlement is the farthest, nor has anyone in real life ever bothered to define that. We've got that deep periphery map from ISP3 as shown above, which is in-universe a
ComStar IE document. No other faction in BattleTech would credibly know more about the deep periphery than ComStar/Word of Blake/IE, though individual factions may know a bit more about their tiny corner of the galaxy.
3. If i shot up a dam will the resulting water rush swept away my enemy's mech, battlearmor and vehicle?
*Inner Sphere Fried Tetatae
The catchwords "dam", "flooding" or "current" don't appear in the TacOps index. Rapids and water flow rules, which are arguably applicable to flooding following a dam breaking, are on pp. 50, 51, 59 of TacOps.
I couldn't find a BattleCorps scenario involving a dam.
In the BattleTech cartoon, there is one scene where Adam Steiner makes the Clan 'Mechs shoot at a dam, then uses his jump jets to evade the flood that sweeps away his enemies. However, the flooding isn't recreated in the scenario for this episode in the 1st Somerset Strikers sourcebook. But at least you have a canon(-ish) instance where something like this apparently happened, given that the 1st Somerset Strikers' exploits are canon at least in broad strokes.