Can mRNA be used by ribosomes more than once?
There are a few cases in translation that I can think of where an mRNA is translated more than once: the closed loop model of mRNA translation, translation reinitiation, and translation by polysomes.

Closed-loop model

Here's an interesting paper to get you started: ncbi 

Essentially, the theory is that mRNA can form a closed loop with the 3' end held to the 5' end by various factors. The ribosome assembles near the start, runs through the mRNA, translating the polypeptide sequence, then dissociates at the STOP codon. Since the "end" of the mRNA is near the "start" of the mRNA, the ribosome can reassemble soon after dissociation.

Translation reinitiation

See this.

The ribosome dissociates incompletely at the STOP codon, the small subunit keeps moving on the mRNA, then the large subunit reattaches at a downstream start site. This happens when the mRNA has multiple ORFs.

Polysomes

(See the wikipedia article on Polysomes for more details)

This basically means that you have a ribosome train moving along an mRNA. The same protein is being translated multiple times, but by different ribosomes. One really cool thing about this is that it's actually been observed by electron microscopy!

(source: bioninja.com.au)   

To answer the 2nd part of your question...
wikipedia

mRNA gets degraded in a process called RNA interference (RNAi). Small interfering RNAs (siRNAs) are also transcribed that are incorporated into a ribonucleoprotein complex (RISC) that has an mRNA cleaving activity. The siRNAs are complementary to segments of mRNAs in order to give specificity to RISC.