Many of the EIFS issues we see are related to improper flashing, etc.  These types of issues can be repaired by removing EIFS in affect area installing PROPER flashing as patching EIFS.  Done by a professional craftsman the repair can be seamless.

You can also remove all of the EIFS and install any kind of material to the house.

We have a client that recently dealt with a similar situation, but it was a racoon. They installed a one-way door, made from galvanized mesh, over the hole. Once the varment was out, the clean-up and repairs could be made. There are liquids that be applied that will keep them away once everything is sealed up.

If you want an efficient fireplace, I recommend a full insert (sealed-combustion) that runs a new vent pipe, typically up your existing chimney. Hire a professional installer from a local fireplace company to do this. 

There are also options to retrofit a "gas log" in an existing fireplace, but there are several issues with this. Typically, if you have an open hearth without doors that seal tightly closed (with a gasket) then you are going to put more energy up the chimney than actually radiates out into the room.

here is a link to a good article on How to get squirrels out of an attic.

http://www.aaanimalcontrol.com/professional-trapper/squirrels-attic.htm

Yes, we almost use Thermory exclusively now. The product is amazing and I really cant think of any negatives as opposed to Ipe that shrinks and is super heavy. We just used Thermory on our own home.

- Sean Sullivan, Living Stone Construction

You can proform both jobs at the same time if you are good at what you do. I have in the past grinded a Grove in the brick chimney all four sides then used a brake to bend a little half inch lip on the top of my flashing metal flashing into massonary Grove then caulked you will never ever have to worry about again in your lifetime Only reason I did massonary work first is because I didn't want to get my shingles all dusted.

What did you decide to do then? Following the proper process would help.

The pictures were very helpful - thank you.  It looks like only the first ply/layer came apart which is not optimal but minimal risk for coming apart allowing the ridge vent to come off especially if they used every nail hole during installation. The bug barrier appears to be intact and the breakage of the plywood would not interfere with that.

Many pests like raccoons, squirrels or mice crawl into the house through the attic and crawl spaces if there are no barriers to prevent them from getting in. They may have mites, fleas, bugs and other pests, which may infest your home. Therefore it is necessary to implant a proper barrier and maintain a proper ventilation in our attics. The pest exterminator New Jersey professionals may also be consulted for the same to get better suggestions.

Our recommendation would be to do the massonry work first and then roofing for a few different reasons.

1.This gives the masons freedom to step and walk on the roof without causing damage to the new shingles. 

2. You will want to make sure the new flashing that is intstalled around the chimney is installed correctly, accounting for any masonry changes and is not damaged due to installation.

3. Ultimately, that the new roof is not damaged and that the warranty does not get voided. 

That would really be a question for the roofer on the split wood.  As for a bug barrier; that would be a question for a pest control company but that has never been a significant issue in our experience.  It is very important that there be adequate ventilation in all attic spaces that are not heated.

Definitely before you do the roof, masonary work and any flashing around the chimney should be done prior to starting on the roofing.

The masonry work should be performed prior to repairs to the existing roofing, unless the roof is leaking, whereupon you may need to make intermediate repairs to stop the leaking while the masonry work is performed.

Then how are you handling this problem? They are ofcourse fraud and justice needs to be done. I think you must reach out to the Home Quality Remodeling contractors online for the professional advice on this. They are licensed and very professional. So, they can solve your problem easily by giving you the best solution possible.