With regards to your edit.
Summary
What you're trying to do doesn't make sense unless you send a few thousand (or more, depending on sensitivity) mapping drones into the galaxy to keep up-to-date mapping data. The drones would be connected in an FTL sneakernet so you can just ping the nearest drones for data from time to time and always be within a few hours of current data.
Mapping within explored solar systems would be done routinely so there will be no issues there.
Without a lot of drones, or very good resolution on them, there will always be uncertainty about the exact positions of small bodies in the unexplored / infrequently visited solar systems. As such, you'll need to send a scout drone ahead to be absolutely certain there's nothing where you're going in you're in unexplored areas.
You're pretty safe blindly jumping anywhere you can see, so it's not that huge of a deal anyways1.
I didn't do any math, but the entire network could potentially be setup within days if your mapping drones can jump hundreds of light years at a time.
Can't take a snapshot from a single point.
There's no way to take a snapshot of the entire galaxy from a single point in the galaxy. The galaxy itself blocks light from reaching us, plus there's a resolution limit due to diffraction. There's (probably) also the pesky super-massive black hole in the center of the galaxy that's difficult to see past.
That said, you could get pretty good pictures of the galaxy by using a number of probes spread through the galaxy. There's no way you'd get exact positions of all the planets, asteroids, and smaller junk from really long distances, but you could at least know where all the major stuff is.
Highly populated areas would have lots of data available for real-time updates on pretty much anything hazardous. Unexplored areas would require mapping before a jump.
Need lots of probes. How many?
Future technology will improve our resolution to an extent, but let's say 1000 light years is a good range to see most stuff (this page says we can use parallax out to a couple thousand, but let's be conservative). Then we just need to cover the galaxy in a 1000 light year grid.
The galaxy is roughly cylindrical. From Wikipedia, it's about 100k light years in diameter, and 1k light years thick. That's a volume of about $2\pi rh$ $=2\pi(50kly)^2\cdot1kly$ $\approx6\cdot10^{13}ly^3$.
In 3D space, the furthest points in a rectangular grid are $\sqrt{d^2+d^2+d^2}$ $=\sqrt{3d^2}$ $=d\sqrt{3}$. This means our grid needs to be $\frac{1000ly}{\sqrt{3}}$ $\approx 577 ly$ from node to node in order for every star to be within 1000 light years of a station.
Think of a bunch of cubic groups. Each group will have eight stations (the corners of the cube). Each cube will be $(577ly)^3$ $=1.92\cdot10^8ly^3$ in volume. That means we need about $\frac{6\cdot10^{13}ly^3}{1.92\cdot10^8ly^3}$ $\approx 312000$ cubes and about $2.5\text{ million}$ stations to blanket the entire galaxy.
That's a lot of stations, but considering you're talking about exploring the galaxy, it's not really that bad. Plus, you only need to blanket the parts of the galaxy you intend to explore.
Modifying the number based on different view distances.
Ok, so we can already see more than 1000 light years using parallax methods. What if you want to calculate a different value?
Well, the number of needed stations is proportional to the size each cube covers, which is proportional to the cube of distance. So cube the ratio of your distance to 1000 light years, and divide the required stations by that. For example, pick 5000 light years.
$\frac{2.5\text{ million stations}}{(\frac{5000ly}{1000ly})^3}$ $=\frac{2.5\text{ million stations}}{125}$ $=20000\text{ stations}$
That's the entire galaxy with only 20 thousand mapping stations.
How do we update that in real-time?
Use the "sneaker net", combined with FTL, like in this question.
Basically, each mapping station has a few small FTL drones that warp back and forth between nearby nodes. A drone's host node gives the drone all its current information via some kind of close-range, wireless (or wired, doesn't really matter) transmission. The drone pops to each "connected" node (the six nearby nodes: up/down, left/right, front/back) and transmits that information to the connected node with the same kind of short-range transmission. At the same time, it would receive information about distance nodes from the connected node. Then the drone pops back to the host node and uploads all the newest data.
You'd need a pretty big dataset to hold all this information, but a society this advanced shouldn't have much trouble with that. Each node has a complete copy of the data at all times. Far away nodes will be slightly out of date, but the FTL nature of the network means they'll be accurate within about $JumpTime\cdot NumberOfNodes$. This is a linear node count; worst case scenario is going to be about $JumpTime\cdot\frac{2\cdot GalaxyDiameter}{NodeDistance}$.
For a 2 minute jump time (time to jump, transfer data, jump back, transfer new data) and 2900 ly node distance (corresponding to a 5000 ly view distance), that's $2min\cdot\frac{200000ly}{2900ly}$ $\approx139min$, or about 2 hours. The number is linear, so if you double the jump time, you'll double the lag time. Also, it depends on how many drones there are per station. The 2 hour number assumes six active drones per station; if you only have one drone popping to 6 stations, that's six times the delay, or about 12 hours.
What about those pesky, unexplored systems?
If you're jumping into uncharted territory, you'll want to send a lead drone. The drone pops in, maps the nearby area, then pops back to the main ship with the results. You just want to make sure you don't hit anything, so the drone can be tiny. If the drone doesn't come back, don't teleport there. Send a second drone a few million miles away and try again.
Space is huge. Even stuff inside the solar system is really far apart. The reality is that you could randomly jump around our solar system for the rest of your life and probably die of old age (or mutiny)1. A couple lead drones should be more than sufficient for most crews.
1Derivation of safety statistic.
The Sun is about 99.8% of the solar system's mass. The Sun's density is $1410 \frac{kg}{m^3}$. Ice comets have a density of $0.6 \frac{g}{cm^3}$ $=600\frac{kg}{m^3}$. The Sun's volume is about $1.4\cdot10^{27}m^3$. Double that and you've got way more than the volume of "stuff" in the solar system. The solar system (just counting out to Pluto) is about 7.5 billion km in radius. That's a volume of about $1.8\cdot10^{30}km^3$.
That means about $\frac{1}{1.3\cdot10^{12}}$ of the solar system is stuff. If you make one jump an hour for 60 years, that's 525600 hours. Your probability per jump of hitting something is $P=\frac{1}{1.3\cdot10^{12}}$. The probability of hitting something after N jumps is $p(n)=1-(1-P)^N$. $p(525600)$ $=1-(1-\frac{1}{1.3\cdot10^{12}})^{525600}$ $\approx 4\cdot10^{-7}$. That means about 1 out of 2.5 million people will hit something if they all jump once an hour for 60 years.