Can a Green Lantern keep replacing injured parts of their body with constructs, so that the Lantern is now made of constructs? Would they still be worthy of the ring?
No. The closest a situation gets to what you are talking about was in the series The Nail, an Elseworlds comic book that has Scot Free (Mr. Miracle) dying under torture by Granny Goodness and the Harpies. A Green Lantern ring happened to be nearby, and Big Barda became a Green Lantern just after Scot died. Scot transferred his consciousness to the Mother Box that Barda wore on her clothing, and then when the Green Lantern ring came, it merged with both the Mother Box and Scot Free's disembodied mind. As a result, Scot "lived" in the power ring, and could make a body completely out of green energy construct. Under adverse conditions Scot "lost" his construct-body and retreated to the Green Lantern ring.
In mainstream DC, the closest situations got to what you are talking about was Soranik Natu, a doctor, making replacement limbs out of green energy. The constructs for some reason wouldn't last beyond Sector Zero, and were very experimental, but Soranik did it.
When an orange lantern, say for example Larfleeze, kills someone he can then create a construct of their identity once they die. These new orange constructs can do they same just like Larfleeze.
There have been cases of when Larfleeze has killed Green Lanterns and reanimated them. However technically they were Green Lanterns that have been remade into orange Lantern constructs.
Hope this helps!