Can a DNS record point to an address like my.domain.com/subdir1
DNS records only map IP addresses to hostnames so in a word, no
You could, however, use a hostname configuration in your web server to serve a subdirectory when a request comes in. Like having something.domain.com redirect/equate to somethingelse.domain.com/downhere. That would depend on your web server software, not DNS.
CNAME records are part of the Domain Name System, the naming system for the internet. Their only purpose is to translate host names, like
www.example.com into IP addresses. You can't attach folder names or queries to a CNAME. To be precise, CNAME resolve into other names, which themselves can be A-Records pointing to IP addresses or other CNAME records.
my.domain.com/subdir1 or better
http://my.domain.com/subdir1 A URI, a Uniform Resource Identifier which is used todentify or name a resource on the Internet - in your case a HTTP resource.
So, a workaround could be to define a CNAME to a different host header, which redirects to your folder using the HTTP status response 301 ("moved permanently"):
-> Browser requests subdir1.domain.com -> Server sends a HTTP status code 301 to the browser: REDIRECT my.domain.com/subdir1
All web servers and web programming languages provide commands for these types of redirects.
What you're asking for is a URL redirect. A CNAME record will NOT accomplish this.
You didn't mention your OS or web server so I'm just trying to provide the best generic answer I can.
No. CNAME records can only point to DNS records. You're looking for a HTTP redirect.