Is your meniscus weaker after part of it has been removed and easier to tear?

Not always. The goal of arthroscopy for treating a meniscus tear is to remove the portion that is causing the symptoms (pain, swelling, & catching). The remaining meniscus is usually stable & less likely to retear because the unstable portion has been removed. A good analogy is trimming a hangnail to avoid propagation of the tear and prevention of further injury (except arthroscopy is performed under water).
Probably. Partial meniscal removal gets rid of the obviously injured part, but often the meniscus has other "micro tears " and or degenerative changes that while not visible at the time of surgery can be a problem in the future. Re-tear requiring further meniscectomy is relatively rare. Often we see re-tears in patients years after their meniscectomy at the time of a total knee replacement.