"I don't watch news that much."
は is a topic particle. The topic of this sentance is the news. を is a direct object particle. You can use it in sentances that describes some kind of action like if you were to say "毎日ニュースを見ます”（I watch the news everyday). In this case は sounds more natural.
No, the verb 見ます requires a direct object. Since you've used が, the subject particle, after ニュース your sentence translates weirdly to "As for me, the news doesn't watch (me) that much"