Can allergens by itself cause fatigue and chest tightness? How good are products like air purifier or nasal filters (woodyknows) for preventing allergy?

Yes. Yes they can cause those symptoms. Air purifier can help clean the air. Nasal filters can help some too. .
YES, can help. Yes, allergens can cause fatigue & asthma. Air filters and ionizers can dramatically reduce allergens in one's indoor environment. I'm not familiar with Woody Knows filters but they look like they could be useful for when one is outdoors and are worth a try. I recommend the book Asthma Survival by Rob Ivker MD.