You can customise your online characters in terms of colours, armor and weapons. However you cannot change the actual person
for example if you unlock a Human warrior you can only customise what He/She wields.
And yes this whole EA server thing is very concerning, Bioware needs to address this ASAP
Sadly I don't think Bioware/EA will address the "server dependency". Please never forget that *what* forced EA to address the Sim City 2013 "always online fiasco scandal lie" was when a "hacker" made a emulation bypass that proved how "little" the EA servers where used to "run" Sim City 2013. As a response down the road the EA PR division had to come up with some sort of "half-baked sell-able lie" to cover their INITIAL LIE, and thus they released a OFFLINE MODE for Sim City 2013, however, do note that the Sim City 2013 "offline emulator" at this stage already was quite operational "for general use".
Depending on the popularity of DA:I we might see a response in the form of a co-op multiplayer emulator just like with Sim City 2013... but that EA willingly will offer a sollution that bypass their "micro transaction" online CHEAT system for players that are willing pay REAL CASH for legitimate cheating... well, dream on!