The synonym list shows "difficult"; why wouldn't an elephant be difficult?
'schwer' can mean 'difficult' but in this particular sentence it normally wouldn't be understood that way. If you want to express that the elephant is 'difficult' (in whatever sense that may be) you can use 'Dieser Elefant ist schwierig'.
The r at the end of Dieser is very silent. I played it twice and still thought it said "Diese Elefant", even though I know that's not grammatically correct.
Sounds fine to me. Note that the quality of the 'e' at the end also changes depending on whether it is 'diese' or 'dieser'
The unaccented -er ending often sounds like -a.
I don't understand why it wouldn't be "Der Elefant"