I was unable to load Opera Mini as well. Same issue as you. I tried Steel though, and I really like it. Dolphin comes recommended in this forum as well. I tried it, but overall, I prefer Steel.
As to your other issue, I have not connected my phone to the computer yet so I cannot comment.
Bit of advice: if you plan to use Nuance voice control, start it, hit menu, go to settings for nuance and train it. That helps a lot.
I traded in a Hero for the Moment and I am glad I did.

 

1. opera mini doesn't work on our phone yet... i'm sure there's a way to get it to run via a java app, but right now it hasn't been running.

2. when you connect your phone to your pc, try power cycling (turning off and on) your phone... sometimes it'll recognize it then. remember, you'll need to 'mount' your SD card from your phone to be able to access your SD card from your computer.

 

As mentioned, Opera doesn't work on the Moment and hasn't been updated in quite a while. Check out Dolphin, it's arguably the best Android browser out there now.
When you try to connect to your pc, are you mounting the sd card on the Moment (pull down the notification menu, select USB connected then mount)?

 

The FAQ has a fix for your connection issue...
http://androidforums.com/samsung-moment/14136-moment-faq-guide-common-questions.html

 

Id definitely try dolphin its my favorite browser ive used i liked steel but i had to many force closes with it so i uninstalled it