>> Can bitcoins be tracked and traced like credit card transactions?
Yes, it can be tracked if the user doesn't take enough measures to protect his/her identity. You increase anonymity by using traditional anonymizing services (tor, proxy, vpn etc.) and then use multiple wallets and multiple addresses to handle your bitcoins. Never use the same address with multiple parties and never re-use the address that you used for transactions which the Government might have an interest to spy on.
>> Can you give your friend a bitcoin worth $200 for fixing your furnace without the government knowing about it?
Yes. Bitcoin, with some very basic pre-cautions, can be easily as safe as cash-only transactions. In practice, it is much safer too.
>> Can anyone pay in bitcoin, "under the table?"
Yes. You just need the reciever's bitcoin address to pay them.
>> Is bitcoin a digital coin that can be tracked and traced anywhere it goes?
Yes the block-chain (the general ledger of transactions) tracks the adresses, bitcoins are sent to and received from.