First thing to check is the USB connector - laptop in a car with usb cable being pulled from time to time, plus the usual vibration will make the connector loose (or with a slack). Apply a very slight pressure on the middle of the connector to create a very small bent on it - that will restore some ''connection stiffness'' with the other connector (at the laptop) and will prevent it from becoming loose again, giving bad contact. Be carefull doing so, too much bent and your connector will be done.
A note about the inverter you are using. If it's the type you plug in a cigarette lighter, then it's not a big power source for your needs. Your laptop alone might work, but when you start plugging things up on it, they are also pulling some juice - and that could lead to low voltage situation and/or overload inverter, or worst... overload your power adapter for your laptop. These are very fragile (weakest link), and cost a lot on some models.
As per my experience, it is a good idea to put some extra $$ to get a more powerfull inverter - you'll get more power (of course), but you will get stability even if your setup is demanding much power. Also, the electrical sine wave is very important - some cheaper inverter are only emulating (step mode) the sine wave - that could disrupt the timing of any electrical port - some laptop power adapter are also very sensitive to the sine wave fluctuation and could stop working just like that - because the step mode is not compatible with that laptop power adapter, i've seen that on recent Dell laptop and some HP.
On the USB side, even if all seems plugged correctly, at some point when you get an under voltage condition, your component can experience out of synch and you'll get issues like you decribed about loosing the contact with your GPS. If you are using a cable adapter (serial to usb), then the problem is doubled because both the cable adapter and the USB port needs to be in synch. If on top of that you have an overloaded inverter with bad sine wave, then it's a sure bet you will experience that situation where you're loosing contact even if all is plugged correctly. Usually you unplug/replug and all starts working again - it just resynchronized all the line from the adapter, to the usb port, to the system... But it will fail again if the inverter is overloaded or giving poor electric sine wave.
On the issue about GPS loosing the satellite signal - i had that weird thing happening to me. It turned out my dash webcam was causing interference (a Sony). I moved both of them further appart and it solve the problem.
