For example, if you take a picture very close to an object, zoomed out, and then move further back and zoom in would the pictures show the same DOF given the aperture was adjusted?
DOF will almost certainly not be the same. You're right to point out that your lens (1:3.5-5.6) will automatically adjust its aperture as you zoom. If you make no other compensation, this will adjust your exposure, so to keep consistent exposure you should also adjust shutter speed or ISO to offset the aperture change invoked by zooming - but that's a side point.
The depth of field is proportional to the distance from the subject. So if you had a lens that had a fixed maximum aperture throughout its zoom range, then under your scenario (stand near, take a shot, step back some way, zoom in, take shot with same framing) - the depth of field will be different. To give you an example with some dummy numbers to give you a sense of what's going on - if you shoot a face from 1 metre at f/2.8, you might have a depth of field that's in focus approximately 1cm from front to back. Now walk back 10 metres, zoom in and take another shot at f/2.8 and your depth of field might now be 5cm or something like that. In the first shot, only the eyes are in focus. In the second shot, so is the nose and so are the ears.
So in your case where the aperture is adjusting, you can almost certainly guarantee the depth of field will not be the same simply because for your second shot the aperture will almost certainly be the same at 10m and 20m but different at 1m. So from 1 to 10m you might fluke it and get the same DOF, but from 10m to 20m you cannot.
so I cannot create a nice DOF
You need to tell us what you consider "nice". Shallow? Or deep?
my kit lens has its sweet spot around f/7.1 and since I want my pictures to be sharp I want to shoot with that aperture
Ok - that's a great starting point. If we know we want that fixed, then we also know from earlier in my answer we can still adjust depth of field by moving toward the subject, or away from it. If you fix your aperture then walk towards and away, find the closest and furthest distances from your subject that make great compositions for you - then inspect the images closely on a PC screen to see the difference in depth of field.
Do I need a new lens with a maximum wider aperture to achieve the outcome I want?
It depends on the outcome you want. A wider aperture will allow shallower depth of focus. A second benefit to wider aperture - even if you stop it down to f/7.1 (although a different lens will have a different sweet spot) is that your auto focus system uses all available light with the aperture wide open. This means in low light your AF will perform better on a lens supporting wider aperture - even if at time of exposure you program it to stop down to a small aperture.
Does focal length really influence DOF? If so, why can't I change DOF by getting closer to the object? Am I doing something wrong?
It's not focal length, per se, but it is distance to subject. The depth of field can in fact be calculated based on your distance from subject - there are websites where you can key in various values and measures to do this, but I don't have a link to hand. Moving closer - while retaining the same aperture - will make your DOF shallower. Perhaps your camera's adjustment of aperture due to zoom is offsetting this. You can test this by taking 2 shots from different distances without adjusting focal length (ie. without using zoom) between shots.
I'm interested in street (especailly at night)/landscape and animal photography
Very cool :) But even knowing your preferred subject matter doesn't tell me what kind of shot you'd like to take :) Hope the above gets you on your way though!