The -r (--refresh) option to omxplayer clears the screen when it resets the video resolution and whatnot.
-r / --refresh adjust framerate/resolution to video
An alias could be useful here:
alias omxplayer='omxplayer -r -o hdmi '
Which will then always refresh (clear) the screen and send audio to the hdmi output. And yes, the trailing space in the alias is intentional as it allows further tab-completion to occur. (Very useful!)
As for the blinky cursor, I rarely (!) ever use a keyboard with my pi's, always logging in remotely (I've built an SD card maker, preconfiguring things), so I added the following into /etc/rc.local:
# turn off console blanking
setterm -blank 0 -cursor off
/etc/rc.local is run as root during boot and this turns off the console blanking which tries to be helpful when no (physical) keyboard activity is sensed, and also turns off the cursor.
Since resetting the video often restores the blinky cursor, I have a script I run to send commands to the console (tty1 actually):
# send 'setterm' commands to /dev/tty1 (physical console)
setterm $* | sudo tee /dev/tty1 > /dev/null
So anytime the cursor reappears, I type:
tty1 -cursor off
and it's gone! Of course, you'll need to set up your user with sudo privileges, at least for the tee command. (look up /etc/sudoers and /etc/sudoers.d) The script could be edited to always turn off the cursor if one wished.
Or just go mad with a combination of these:
alias omxplayer='tty1 -cursor off; omxplayer -r -o hdmi '
Which would ensure the cursor is off, then start omxplayer. Play with it!