F5 VE LTM Trial to Production

Is it possible to start deploying the F5 VE LTM evaluation first? 

do you mean the trial version download or with a trial key from an F5 employee? the download trial is quite different, that probably won't be able to be licensed normally.

Once we got the license, can I just input the license key without affecting the configuration?

not 100% sure, from past questions we were told it wasn't possible to "upgrade" a lower Mbps VE license to a higher one. perhaps a trial license is exception here. but in any case you could create a config backup and restore it on a new instance without the license so it takes some more work but you config isn't lost.

Is it possible to get the F5 LTM version 12 ova file ?

yes it is if you have access to the F5 downloads site.