Is the only way to get to Lothlorien "through" the mines? Will there be any other way to get there?
At launch, all roads east of the Misty Mountains lead through Moria. Other paths may open (such as the High Pass or Red Horn Pass) not for Moria launch.
How would stable rides work then? Will horses travel through the mines?
For launch, there is no travel system associated with Lothlorien. The Elves of that land are suspicious of outsiders and do not allow outsiders to travel freely in their lands. The Travel System in Moria is its own bubble for the time being. You'll enter from Eregion (whose travel system is integrated with the rest of Eriador) and from there be able to access routes to take you around Moria.
Be aware - due to our desire to have a stable Moria experience at launch, the number of swift travel routes available is limited. It will expand in the future, but we like keeping our servers as happy as possible under the crush of all the players we expect to be flooding into Moria.
The Lord of the Rings Online Team