The role is very dependent on where you work and what they think a DBA should do. If you really want to write SQL code for apps you should look for SQL Dev roles.
A general DBA role is more focused on the backend stuff, keep everything from falling apart, writing automation, making sure the automation is working, performance monitoring and troubleshooting, on call to fix shit if it breaks in the middle of the night. In general any sql code I write is specifically for stuff I need to automate/get done dynamically.
Some places have DBAs more involved in the app side, some have good performance dba's who'll they'll have check all the sql code (that's unsustainable a lot of places though due to usually staffing a low number of DBAs vs. the amount of code that gets written/modified).
I can't really comment on pay, I've seen it vary widely and I don't know dev salaries at all. I would guess you make more money to start as a dev because DBA roles are really looking for people with experience and a good track record since at the end of the day you are responsible for production and DR and all that. But with some years under your belt you could probably end up making more, especially if you go towards consulting and that kind of world, which as a DBA is more accessible because its more specialized.