A major problem I am thinking that you may have is the fact that live algae clogs DE filters up almost immediately. You are best to shock the pool and then put your filter on recirculate until the algae is dead. I don't know about "damaged skeletons" but maybe someone else will chime in here who does.

A couple of other things that I am noticing as maybe contributing to your problem:
-- pool is receiving maintenance only 1 or 2 times per week. (When you are fighting algae, it is everyday battle or else the algae wins.)
-- adding algaecides can cause issues. The only algaecide we recommend is 60% Polyquat.

We will need some good numbers taken with a drops-based kit. If you don't have a good drops-based test kit, you need to get one.
The test kit that we think is by far the best is the Taylor K2006 or 2006C (better buy). Not available locally but you can get it through this link that takes you to Amazon: http://pool9.net/tk/

But, in the meantime, go to Walmart and see if they have the HTH 6-Way drops-based kit (no test strips!) and get that. If they don't, get an OTO/Phenol Red kit (yellow and red drops) instead. Use it to test and then post your numbers. Also while you are at WallyWorld, get several jugs of their generic 8.25% bleach and a couple of boxes of 20 Mule Team Borax (laundry aisle).

We also need to know exactly what all has been added to the pool meaning ingredients and not just product names like 'shock' or 'algaecide.'

(One other thing that is likely to be problematic for us in helping with your pool is that it is difficult for us to offer advice to a homeowner when there is a pool company also involved. Mixing and matching our advice along with what they may or may not be doing usually doesn't work.)

Welcome to the Pool Forum!