Crossing into the West Bank itself is simply a matter of continuing on whatever road you were already on and passing a check point, (though usually they are only checking anyone going the other way) and it's perfectly doable with an Israeli taxi, or any one else willing to take you there for that matter.
Furthermore, it's just as doable to procede from there into an Israeli settlement; it's just like driving into any other town.
However, I would recommend finding out the best way to get there in advance from the person who lives there, as many roads can be much more safe than others.
It is also perfectly possible to return to the fully recognized part of Israel, this time through an actual checkpoint, again in any Israeli vehicle. Whether you will have to pre -arrange depends on where you are; your host will likely know local taxi options. However, for a journey to Tel Aviv it may be a good idea to pre-book regardless.
In terms of general safety, both day and night it depends where exactly you are. But in all cases most likely people living there will say it's safe enough for them to travel. At the same time there are other places that are probably statistically safer to be, so it really is up to your comfort level. This is another question that would be best for your host who is likely familiar with the exact roads you will be on.
(Also, obviously a taxi from Tel Aviv will cost quite a bit)
Getting to Palestinian towns from the west bank will be very different, as Israeli taxis will not go there, and there may be more security checks on the way in and/or out of the towns/cities. You will have to take an Israeli car into the west bank as above and then transfer somewhere to Palestinian transportation.