The BB browser is one of the applications that is on all the time. Hold alt +escape to check which apps are running. Should be 5 full-time, browser,home,messenger,messages, and call logs.
I don't have the answer to the second question.
Posted from my CrackBerry at wapforums.crackberry.com
What bank do you use? Just because its not in 1 app doesn't mean its not in another application. You need to figure out what OS you have by going into your options and clicking "about" if you're at 4.5+ then you're fine on installing almost all applications.
Do a google search or look around a couple blackberry websites to see what you can find to support your online mobile banking problem.
