There are DNS hosting services that let you set records for your domain. For example you can set A record to point to your webhost IP.
But most of the webhosts also allow you to set A records.
Would the DNS lookups be faster if I used the webhost directly?
There’s really no way to know without actually comparing. But the usual opinion is that DNS hosting services compete based on how well they can host DNS while web hosting companies often just provide DNS hosting services in case people need them or to provide one stop shopping. DNS hosting services have to prioritize the quality of their DNS hosting while web hosting companies don’t. Nevertheless, of course, a high-end web hosting company may well provide faster DNS hosting than a low-end DNS hosting service.