I myself use LastPass which, by way of a web-browser extension, directly fills login form fields by way of identifying the field DOM IDs and directly setting their value to the appropriate information, thus circumventing any sort of copy + paste business. If LastPass is unable to identify the login fields automatically you may manually inspect the form and enter the field IDs into the LastPass profile by hand.
All of my website passwords these days are unique 20+ random character strings, but thanks to LastPass I'm automatically logged in to sites with the service's "Autologin" feature as soon as I visit the site. I hardly ever even click "login" or "submit" anymore.
For logging into the service itself, you are provided with the option to use an on-screen keyboard to thwart any efforts of keyloggers that may have found their way onto your system. Other (not entirely relevant) security features include one-time passwords and two-factor authentication through a variety of different auth services.
On the mobile side of things, the LastPass app for Android supplies a lightweight web-browser that fills in form fields for websites with saved information, once again circumventing any use of the clipboard and copy + paste procedures.
As you implied, LastPass also supplies an alternate keyboard allowing for credential insertion into most Android apps and web-browsers.
This of course does not work for credentials for programs or other secured non-website/app items, all of which still require that I copy and paste credentials as needed.
There is also the possibility of selecting the text of your usernames/passwords and click + dragging them into the form fields which may circumvent the system clipboard depending on your OS.