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2. Yes it is possible. Recommended or not depends on the client, the thinner the better (Like Electrum). The problems that might arise is the wallet arrangement of keys algo might get confused. In some cases people lost bitcoins because of automatically generated change address which typically were not present at the other computer.
3.The source can always be seen, Hence alot of devs here like gmaxwell started projects like zerocoin. (did I get the name right?)
In regards to number 2, you do not want to run the same address on multiple computers. Things can get messed up. Change addresses are different between the multiple machines for instance. If you really must have wallets on multiple machines, point them all to a single data directory that is available to them... like if you've got some NAS device. Backup your wallet.dat frequently (USB sticks, external drives, etc).
What is the change address actually? So the client transfers your bitcoins from one or more of your receiving addresses to the change address (which is automatically and randomly generated?) when a transaction happens?
My question/problem was basically about a situation if one has 2 computers and wants to use bitcoins on both of them. If he receives bitcoins on computer A (let's say it's a desktop computer) then goes travelling and must pay using his laptop (comp B), what is the best solution?
3. Are you referring to the addresses? Like 1askjdwqkjeljqwkDasdaeb4 sent x amount of btc to 1daoeiwqueioquw3h2? Or are you referring to the location of the sender? Confused. 
3. By "source of the transaction", they mean the IP address.
I was basically referring to the bitcoin address. So it can be seen even though Bitcoin Core client doesn't show it.?
