Sir, in this modern age, is it ever proper to put down the pen and resort to e-mail? When many have given to owning only a cell phone, how dare we call them just to leave a message? Being quite old and residing in an old folks home, I'm shocked at what some of my neighbors consider proper. For those in younger years, just you wait for old age waits for no one.
Dear Corresponding Regardless:
No matter one's age, Speed has overtaken Graciousness in communications, especially in the last 25 years with the universal adoption of cellphones and email. A well-turned phrase and a well-rounded period rarely appear to advantage via text message. But humans adapt to changes in technology. We aren't, for instance, still scratching with styli on clay tablets.
To answer your first question, resort to email when expedience matters. Email has become so ubiquitous that it has become Proper. (How one uses it determines whether it is Proper or Perfectly Proper.) When the speed of your communication doesn't matter, by all means write a Proper Letter.
If you are eager to communicate with someone who has only given you a phone number, then you must use the phone number to make a phone call, or discover by whatever means you have at your disposal what their mailing address is and send a Proper Letter.
You refer to living in "an old folks home," which leads Etiquetteer to observe that yes, there are many senior citizens who have not embraced the Digital Revolution of the last 25 years. Whether through Fear, Skepticism, General Cantankerousness, or even Lack of Equipment, they miss out on keeping up with family and friends. (And how many grandparents do we know who finally get on Facebook to find out what their grandchildren are doing, only for the grandkids to abandon Facebook for Instagram or Snapchat or Something-or-Other.)*
At a time when others should be making a special effort to reach out to them, these senior citizens find themselves making special efforts to reach out in the ways they know (mostly written correspondence and phone calls). Besides which, current technology is not always designed to accommodate the elderly. Large, unsteady fingers obscure closely-set buttons so that one doesn't always know what button one is pressing. Designers, take note.
With National Card and Letter-Writing Month coming next month (so designated by the United States Postal Service), Etiquetteer hopes that you will make more special efforts than usual to communicate with the written word.
*Even Etiquetteer, who has yet to achieve the age of "Get off my lawn," has had to learn to text, but it was a hard-won battle.