Grab a coffee or other beverage, this is a long one. 
In addition to years of video production, for a time I used to run a lab with over 40 MiniDV camcorders, mostly various versions of the Canon ZR series.
Normal wear and tear on camcorders should only be to the moving parts. Those would, of course, include all the bits that move and guide the tape back and forth within the body of the camera, but also the eject mechanism and zoom and focus mechanisms.
What we found to be the major break-down, though, was the FireWire connector itself.
Although it shouldn't be a problem, and doesn't seem to be on the Sony cameras we used, several of the Canon camcorders came up with non-working FireWire connectors. It turns out that you shouldn't connect and disconnect it when the camera is powered up. Also, since it is soldered to the main board inside the camera, try to not wiggle the cord when attaching or removing it, as well as try to make sure it doesn't get pulled to a side, or bumped once connected. The repairs would have been almost as much as a new camera, so we used those only for shooting and playing back to a monitor or copying to or from VHS with the analog audio and video cable.
Of course, those cameras were used and abused by many students, at least some of whom were not too concerned about taking care of equipment they didn't own. 
Once connected and (then) turned on, the camera would have only the same wear as your stereo system or LCD TV or monitor. All electronics will age and break down over time, but not so you'd notice, except for possibly contact corrosion with switches and push buttons or rotary dials for volume, selecting modes or menu options, such as the on/off switch, as well as the electric zoom control and on some models, manual zoom and focus as well.
To make a long story short, converting VHS tapes won't harm it any more than just having it turned on.
What format to store the footage depends on what you intend to do with it. If it is just going to be viewed, DVDs will work. If you intend to use it for editing, it is not recommended due to the quality loss during compression to MPEG-2, which is the DVD format. Some people just archive original tapes, in this case MiniDV copies, but with hard drives getting quite inexpensive, many now just save them to external drives.
I would suggest one for the video anyway, as it takes up quite a bit of room on your hard drive, and in any case, it is more efficient to have the operating system and editing software on a different drive than the data. A second drive for back-up would be ideal, as well as a UPS system for protection against power outages or even brown-outs, which can corrupt a hard drive. I had one fry with a power outage (even though I had a surge protector), before I bought a UPS with enough protected AC outlets to include the two external drives I had at the time. (I have three now.)
I would recommend a FireWire hard drive. Most if not all have two FireWire 400 connectors, so you can plug the hard drive into the computer, and the camera into the drive directly. Although USB2 is rated at 480 Mb/s compared 400 Mb/s for FireWire, the FireWire drive will be considerably faster at continuous throughput such as video, whereas with the USB drive, there is the risk of dropped video frames, making the video stutter on playback. I won't go into the reasons here, but here is a link that explains why USB is slower than it's advertised:
http://www.cwol.com/firewire/firewire-vs-usb.htm
If you do want to store video tape, VHS or digital, it will have only a limited shelf life, so you need to store them in a cool place, low humidity, stored after running it fast forward. Rewind and fast forward them again annually, and store the cassettes on edge, not flat. Storing them flat puts the weight of the tape pack on one side, possibly causing edge damage and then the tape may not track properly in the camera or VCR. Wider tape is at more risk of this, having more weight, but why risk it with narrower tapes?
Periodically, 5 to 10 years maximum, you should copy tapes or risk damage due to physical breakdown of the magnetic layer or the glue that holds it to the tape backing. This allows them to be copied to new, more modern formats if they become available in the meantime, as well, which protects you from having tapes for which you can't find working players. Hard drives also have life spans, but usually you can tell when they near their end of life, but for longterm security, these should also be copied to new drives after 5 years or so.
I hope you didn't get too confused with all of the above, but I wanted to give you all the information you may need to make your decisions.