I'm sorry to hear about your recent diagnosis. I've been a long-time follower of letsrun but typically don't post. I can't imagine being diagnosed with cancer at the height of fitness. I'm currently finishing my oncology training and still run competitively (90-110 mi/week).
Going into any cancer treatment as fit as possible has always been shown to improve tolerability to treatment and survival after diagnosis. It doesn't mean that the cancer will respond any better or worse to treatment, it just means that you can take stronger treatments (chemotherapy, radiation) should they be recommended. Given your age and current level of fitness, it is more likely that your oncologists will be much more aggressive with treatments than they would for a 60 year old guy who has smoked for 40 years.
You mentioned you had a genetic mutation associated with your cancer. If by chance your cancer was associated with something called an "ALK" mutation, then having this mutation does allow for targetable treatment. ALK mutation-associated lung cancer is much more common in young patients who have no history of smoking (I assume you don't smoke being a runner). The outcome (survival time, time to progression) of metastatic lung cancer with an ALK mutation is dramatically improved when treated with targeted therapies (like crizotinib). These targeting drugs are tolerated very well and much better than standard chemotherapy regimens. Chemotherapy itself also has dramatically different rates of tolerability depending upon the disease being treated. As the physicians you've seen have suggested, the current paradigm for Stage IV cancer, especially of the lung, is indeed incurable. With that being said, mutation associated metastatic lung cancer in the modern era is being treated more as a chronic disease (given years of survival typically seen). As with any chronic disease, there is going to be close follow-up and as long as the malignancy is controlled, therapy will continue. Should there be progression, they'll likely change course and try another strategy.
Another promising recent development in lung cancer, especially metastatic, is in the realm of immunotherapy. These treatments improve the bodies immune system in combating the cancer. It has quickly become an early recommended treatment for many cancers, not just of the lung. Immunotherapy is just one example of how medicine is developing newer treatments all the time.
So long as you continue to feel well, continuing to run shouldn't affect your response to treatment. If anything, it will help keep you fit for future treatments. I wouldn't overdo the training but running for basic fitness will keep you feeling well in general and some semblance of normalcy to life. It will also improve your eligibility for trial enrollment should standard therapies no longer work.
I hope this bit helps give you some clarity. If you have other question, I'll try to answer the best I can.