Goats are not too difficult to unload, as they will happily (sorry for using that word) jump out from the trailer/van and walk in with you leading them.
I have to say that I have absolutely no problem taking in sheep or pigs, but chicken out with goats every time and get the OH to do it... maybe if you had someone else to come with you at least (to drive home for when you will be upset) it would be less stressful for you.
but I have no problems in cutting up the carcass, we have done ours now a few times.
Best to ask if they do goats regularly and maybe bring him on a day when there are lots of other goats about.
Also have your butcher sorted (for him to collect the carcass and cut up as you want it), so you only have to pick up ready packed stuff. - Burgers are good and lots of cubed meat for curries and other slow stews.