Showing their name in your ad may be a different story. They can tell Google they want their name trademarked and not showing in ads. I don't believe they can do anything about you having the name in your ad until they do so. It would be a good idea until then to have the trademark symbol in your ad next to the name.
Any advice on what the current legalities are with this scenario?
No such advice can be found here - simply because nobody here knows much about your situation.
Depending on the real scenario, it may be nothing, something, or everything.
You're the only one here knowing how worth is what's happening out there - my understanding is that you're actually bidding on their name and want it to stay like that, and that you're compliant with what Google enforces out there.
Some companies are very protective to their TM(s) and they will always take things to the highest level (like court) and "kill" the opponent by the cost, even if they don't win at the very end.
Your best bet is to seek legal advice - if it's that serious and you don't want to give up on their name. ;)
Otherwise, stop bidding, or enter it as a negative match if it's about broad match causing your ad to appear - end of story.
And yeah, if it's some kind of a general word or phrase, then you should have some ground for the behavior of your ad(s), but still, check if you can get an advice from somebody who's qualified.
My ad does not use their trademark at all.
You may not use it in the foreground but do you use it in the background or in your keyword list?
Was this letter from an attorney?
Do you think Google will sit beside you in a courtroom to help you or quickly divorce itself of any legal entanglements if subpoena for evidence is made? I know most would quickly hand over the evidence in a heartbeat to divorce themselves from culpability especially with a total stranger.
Of course this is my opinion and is in no way legal advise.... However before you entertain their letter with an attorney, I would do a couple things.
1) Ask them to provide the registration for the trademark.
2) Ask them to detail where and how you are infringing on their trademark.
3) Ask them to provide examples of your infringement. For example a screen shot of your ad using their trademark.
Usually a cease and desist letter is a empty threat. Most often only someone with a lot to lose and and deep pockets will pursue this in court.
They have a tough case to prove on their hands.
My understanding of Trademark infringement hinges on "confusion in the marketplace", also from what I have read you actually have to use the mark, which you are saying you aren't.
I have had 4 cease and desist letters in the past 9 years. I only replied to one and I asked for proof of infringement and I never heard back, the other 3 I ignored as they were as ridiculous as the one you got and I never heard another thing about them.
I wouldn't worry about it until an actual lawyer contacts you. A cease and desist letter cost $50 to get and you don't have to be in the legal right to send one.
If googling the trademark returns many results including your ad I am not sure how they could say your ad causes confusion when there is likely 100,000+ results.
We had a similar situation a few years ago when we received such a letter from a competitor. The funny thing was THEY were bidding on our brand name too. We took the doc, did a find and replace using their company name and sent it right back to them!
Cooler heads prevailed and we came to an agreement that we would not bid on each others brand terms.