2 questions

1 - In my experience, Opera mini is pretty fast, but there are other good ones too. Some do some things well but may not have some features you want. It's probably best for you to try a few out and see which is best for you.

Some others you may want to try are:
Dolphin & Dolphin mini
Boat & Boat mini
Maxthon
Firefox
Ninesky

2- Sounds like you've downloaded an app that uses "Push Notifications", such as Airpush, Leadbolt etc. See this thread for more info:
http://androidforums.com/android-lounge/325803-seeing-ads-my-notification-bar.html

 

Uninstall the task killer. Totality uneeded and counter-productive.

Install Air Push Detector. It will tell you which is the offending app.

 

right off the bat I can tell you that Music Junk uses AirPush........ and will give you the green star that suggests a 'market update' but actually sends you to an online ad site

not sure about the rest on your list but I know that one first hand

 

For pure speed, I found Opera & Ninesky to be fastest for me. Of course YMMV.

Download addonsdetector and/or airpushdetector to help you identify which app is pushing these ads to you.

 

Try Mirin browser too. I've used it happily for a while, but now I'm on Maxthon Mobile & ready liking it

 

The advantage to Opera is that they compress images on their servers before it hits your device, do you trade off image clarity (or disable them altogether) in order to get text delivered faster.

 

http://androidforums.com/android-applications/335110-why-you-dont-need-task-killer.html#post2681361

 

Read this as to why you should consider removing your task killers: In light of all the discussions right now about why NOT to use a task killer, from a technical standpoint here's what that submitter got right, what they got wrong, and what you should actually be doing to maximize your phone's potential. : Android