Lets consider the following situation:
- you have a Linux (Ubuntu, Mint, Debian) with a Gnome3 desktop with the default Gnome network manager application
- you install tor and restart the machine (by default it start automatically at every startup) 127.0.0.1:9050
- you setup a PPTN VPN using the network manager
- you connect to the VPN using the network manager GUI
- after this you "send a packet through TOR" (e.g. using a TOR browser bundle, using proxychains to route, or writing your own program)

The question is: where the packet goes? Does it go through the VPN and then to the first, second etc TOR node, and then the TOR exit node? I don't know the non-graphical tools like route, etc. to find this out so please help me.

VPN operates on a lower layer, so packets go first through the vpn, then through tor-entry, tor-middle and tor-exit node to the server. The tor entry node does not see your ip, but your vpn-exit's ip.

Is it possible to trace back TOR traffic to the user? (consider the situation mentioned before)
- is it possible to do it when you have "spies" at tor-entry and at tor-exit?
- is it possible to do so "without spying" (eavesdropping) in a normal situation? (consider that you have all the resources needed)

Does the VPN add extra privacy/anonimity or doesn't? (think of an anonymous VPN where there are no user billing informations)

Another "foolish" question: if Alice and Bob are both connected to a VPN (PPTP), how do the "outer world" sees them? Do they have the same IP (like when I have a router at home, I connect to the Internet using that and all my PCs, phones and tables look like they had the same IP on the Internet) or do they have "separate IPs"?

there is NO SIMPLE ANSWER !!!!!

so the answer is
YES!!!!!!
but with some if's and maybe's and kind of's


yes
.onion sites are actually EASIER to trace back to YOU

but
with a bunch of
"maybe's"
"they might's"
"sometimes"
and
"SOMETIMES NOT!!! "


please READ!!!! and STUDY!!!!
the links provided on the tor site
and from security researchers they link to

https://www.torproject.org/

basically
Do your own research

you did learn how to write a research paper in school ? right ?

First, thank you for the answer.
I have already studied some forums and other sources.
I know one very "easy" way to trace back Tor: if somebody owns a big amount of exit-relays or somebody is spying at both ends of the network. These kind of things mean (from my point of view) that *top level* criminals or non-criminals can be observed by big player adversaries like NSA.
I will read more Tor project related papers I just wanted to see some examples to those maybe's and sometimes.

"yes, .onion sites are actually EASIER to trace back to YOU" I think you meant that if you visit hidden services in this case it is easier to trace you back. In my question, I thought of a usual case: what if a user visits a legal site and some people want to trace it back (some people can mean someone like my ISP or normal police, but not CIA, NSA, Interpol, Google or Microsoft and not my neighbour/friend)
Thank you.