You need to hire a trademark attorney to research and then evaluate the facts to determine if the mark has, as a legal matter, actually been abandoned. Not using the mark is not abandonment. Until that's done no one can -- or should -- provide you with a roadmap on how to respond to the company's trademark registration applications. In any event, you have very little to no chance of responding on your own. Only a trademark attorney has the knowledge and experience to do so. You should not even write the company a letter -- even to open a dialogue about potential buying the mark.
The above is general information ONLY and is not legal advice, does not form an attorney-client relationship, and should NOT be relied upon to take or refrain from taking any action. I am not your attorney. You should seek the advice of competent counsel before taking any action related to your inquiry.
You should talk with a trademark attorney. Even though you've done a nice job expressing the nature of your problem, trademark law is highly fact-dependent and a trademark attorney will need to ask you some follow-up questions, investigate and review the filings you mention, and consider available strategies in relation to your business goals and budget for legal services.
You need to review the situation with trademark counsel to ascertain whether you have a claim that this trademark has been abandoned, or whether the trademark owner has engaged in misrepresentations to the trademark office which might give you a basis for challenging the registration of this trademark. You should not assume that merely because a company is not using the trademark in commerce that it has abandoned it, nor should you be quick to conclude that this company is engaging in trademark abuse. There may be many valid reasons for continuing to maintain a registered trademark in circumstances such as this.
My guess is that you are mistakenly focussing to much attention on registration, rather than the issue of whether your use of the trademark in commerce and/or registration of the trademark might be appropriate. Trademark rights arise primarily from use in commerce, not registration. There mere fact that someone owns a registered trademark does not mean that no one else can use that same trademark. It is possible that two companies can use the same trademark as long as there is no likelihood of consumer confusion. Trademarks are registered in one or more "classes" and if your business does not compete in the same industry for the same class of products and/or customers, it is quite possible that you can legitimately use and register the trademark in your business. In a situation like this, it is critical for you to retain and work with experienced intellectual property counsel---otherwise you will continue to make various obviously incorrect legal assumptions.
It looks like you should talk to a trademark attorney. At least that's what 9 out of 10 trademark attorneys would say. I actually agree lest I be condemned for making light of the fact that most of the answers to trademark questions must contain this statement. Assuming (that's a strong word) that the company HAS abandoned the mark, you have nothing to worry about. Their filings are doing nothing but perhaps preventing you from registering your mark. If that's what you want to do then...I won't say it.