There is indirect PvP aspects in Planetary Interaction, and soon there will be direct PvP aspects with the release of a console game called DUST514.
The indirect PvP aspect is fighting over resources. This was only turned on in some of the later patches in the Incursion. On each planet there is only a limited number of the resources in the ground, which only refill at a certain static rate. The planet will support some (unspecified) minimum amount of resource being extracted, then after that it will deteriorate. This means that too much concentration in the area will deplete the resource for all players pulling that resource out of the ground at that place.
The direct PvP will come later when the DUST514 console shooter game comes out. Exact details haven't been confirmed yet, but the base premise of it is that people playing Eve Online will be able to somehow ask players in DUST514 to somehow disrupt other Eve Online players Planetary Interaction networks.
There are usually two types of Planetary Interaction (PI) setups : Extraction and Factory setups.
With Extraction setups, PI doesn't really require any sort of ongoing ISK costs to run besides export fees of planetary products, therefore all PI here should be "profitable". In High-Sec you won't get very high extraction rates at all, and they will be highly contested, which means not very profitable on a "ISK-per-time-invested" basis. Lo-Sec can be better, but Null-Sec and Wormhole planets can be quite lucrative. There is a larger risk in transporting the PI products from these lower security areas of space though.
With Factory Setups, people can make a fair amount of ISK by making PI setups in High Sec that purely concentrate on turning lower level cheaper PI products bought from the market into higher level more expensive products. You'd need to figure out what is profitable here and what isn't as it would also be quite easy to lose a lot of money doing this if you chose the wrong products, or the sell prices of the high-end product dropped unexpectedly.
In terms of investment of training time, it doesn't take too long to get decent PI skills. Anywhere between a few days for some basic skills to a couple of months to be able to run multiple planets with high level command centres depending on how much PI you want to be involved in.
EvE is a purely PvP game with minimal PvE elements mixed in for money making. It is possible to sit in a station and just trade all day, or do planetary interaction or other extremely dull aspects of the game - but the game mostly revolves around spaceships fighting other spaceships.
There are missions with NPC pirates and space-rocks to fight (mine), however the missions aren't very deep and really there isn't enough content in PvE to enjoy. The game is PvP-oriented in that your internet space ship should be fighting other player's internet spaceships.
Think of it like chess. You can move your pieces around the board for no reason, but ultimately to play the game you need to attack your opponent. It just doesn't make sense to not play competitively.
edit: I would like to add that, there is some interesting PvE content such as group mining, corporation management and wormhole exploration - but all of these come with some PvP aspect to it.
Best Answer
This is actually a bit complicated, depending on the scenario.
What happens when you log back on? In case #1, your ship will still be in place if your enemy hasn't destroyed it yet. In cases #2, #3 and #5 where your ship has performed an emergency warp, it will reappear in space at the emergency warp location and automatically warp back to the location you were at when logging off. This is assuming that the enemy has not probed you down in case #2 and destroyed your ship.
In case #4, your ship will actually appear at some random point in space a couple AU from the POS and automatically warp back to the location in the POS from which you logged off. Note that in case #4, your ship will actually warp through any warp disruption bubbles that may or may not have been erected around your POS when you logged off. Again, the assumption is that inside your POS shields your ship should be safe, so you should not be able to lose your ship just because you logged off at an in opportune time.
In EVE, nothing is simple, but I think this covers just about any log off scenario and the various game mechanics involved.