Trample says it works like the Overrun ability so I believe it must be in a straight line. However, it also says that an opponent cannot be damaged more than once per round no matter how many times your movement takes you over the target creature so that makes me think you can travel multiple directions in one go.
I would say no to the Overrun feats since Trample is a special attack that functions like Overrun but isn't Overrun.
Trample uses Slam damage so I think Inspire Courage would apply to a Trample. Greater Magic Fang would certainly work but it can be tricky. If the creature already has a Slam attack then you're good to buff that. If it does NOT have a Slam attack, you must apply the +1 to all natural attacks for their Trample to gain a benefit.
The Barometz has the overrun feats but does not have an increased DC on it's trample. I checked a few other monsters and this is constant.
After going through the bestiary, some monsters do not apply their slam damage abilities to their trample, some apply similar but different abilities, some apply the same ability their slam has, and some add whole new abilities.
This implies that your melee special bonuses do not apply to the slam unless specially stated, heck the monsters don't even have a size modifier added to the DC. That is a strong argument that other effects that modify attack rolls will not modify the DC.
So the wild shaped mastodon druid will not be able to hit everyone he tramples with his phase locking amulet of mighty fists.