Is Lagrange point L1 stable? If I were to place a space station in L1 will it remain in orbit without any difficulty? If there are any difficulties, please mention them here along with any suggestions to overcome the difficulties, if possible.
Something parked at L1 is balanced between evenly matched tug-of-war teams.
On one side of the tug-of-war is the moon's gravity and inertia in a rotating frame (what we used to call centrifugal force). On the other side is earth's gravity.
Go a little bit to the left and Earth has the advantage. Move a little bit to the right and the moon and so called centrifugal force get stronger.
So just a tiny nudge destroys this delicate balance. Something at the Earth Moon L1 (EML1) needs station keeping.
If you want something high up that is long lasting and maintenance free, I would suggest a lunar Distant Retrograde Orbit (DRO).
An example of a moon in a DRO is Jupiter's S/2003 J2. According to the Wikipedia article:
Retrograde moons with axes up to 67% of the Hill radius are believed to be stable.
The moon's Hill radius is about 60,000 kilometers. So a lunar retrograde orbit at 40,000 kilometers would be pretty stable.
And in fact a lunar DRO is one of the suggested destinations for the proposed asteroid retrieval mission.