The thing that is curved is spacetime so drawing pictures of space doesn't necessarily help. And furthermore since it is the curvature of spacetime, a 4d object, there isn't an obvious connect between the length of a drawing and the actual geometry of the spacetime.
So what happens is you have a blob of 4d spacetime and you have to say for each curved how long a ruler would say it is or how long a clock would say it is (for two totally different meanings of long).
So for instance people might try to imagine the space outside a star as curved like a funnel, but that firstly ignores the curvature of time and secondly imagines a larger dimension in which it curves. In reality you have a blob and its like a ruler is smaller when pointing towards or away from the star than when pointing orthogonal to the direction towards/away from the star. And the clocks tick slower when closer.
Now you could also have them be bigger when orthogonal and tick faster when farther. And you might only be able to do this one blob at a time and need multiple blobs. But you basically figure out how actual distances and durations are related to which parts of the blob are connected by a path.
So now it is easy to see how space expands behind you, it's like the rulers are just smaller there and it is easy to see how space contracts in front of you, it's like the rulers get bigger there. You don't need another dimension, you just need to know how to get a distance or a duration between two points in the 4d blob along a path.
Each point in the 4d blob is a time and a location.
But one of my question, for the observer outside the Warped region the time/distance taken by some one travelling over the Warp would be similar to the time/distance before the Warp? Wouldn't?. Cause the External observer is outside the gravitational field and was unaffected, His clock would tick at regular phase, from his prespective the person riding the warp would look like he was strecthed and slowed and reached the final desitination at same time as it took before the warp, or Am I missing something?
Your language seems to not even allow you to hear the physics or give someone room to say correct physics. So let's talk about Newtonian physics. In Newtonian physics, there is TIME, absolute time, and basically everyone, regardless of whether they are standing near a massive body or regardless of how they move can measure the change in TIME with a clock and they all measure the same thing. Does the universe have to be like that? No. Is it like that? No. If you talk and listen in a way where you don't allow the universe to be different than that, then will you be unable to understand the way the universe is? Yes, you will be unable to understand the universe.
People outside a spherical shell of mass experience time dilation too. I made the shell to save you fuel because you can sit inside the shell and get the time dilation without having to fire rockets while you are inside.
But your question is also so vague as to be hard to answer, again because you act like there is some absolute time and that it makes sense to talk about it. In physics, time is something measured by a clock. It is is an interval, a duration, so you measure from one event (a when-where) to another event (another when-where) along a path (a whole bunch of when-wheres in between). And what you measure depends on the exact path. Just like of you used string to measure distances between two points then the answer you get depends on your path. Except you can imagine that if your string was a metal chain with links and if the region between the two points had a temperature field that makes the chain bigger in hot regions and smaller in colder regions then if you wanted the shortest path you'd have to take that temperature into account.
How is the vagueness a problem? Firstly, a person can't tell how fast they are going, so you 40mph versus 80mph dozen make sense because we don't know who is asking about the two things. You can feel yourself fire a rocket and you can see the color of stars change and when you see both happen at the same time you can hope it was caused by you. But special relativity measures less time between two events if you take a path that involves rockets. So not using a rocket does not make you feel like you are going faster since you age a lot.
If what you want is a warp drive, then you need exotic matter, which is rarer than antimatter. And that's an understatement. So the word exotic means truly exotic. But let's say you have some. And let's say that you've been sending care packages between here and Alpha Centari and Earth for centuries. And let's say you send them at almost the speed of light each way, so you send a package and wait nine years and get a response. With a warp drive you could get an answer back in 5 years, something impossible without a warp drive. With multiple warp drives you could do more. It seems like you could go back into your own past..
Which does not bode well for finding exotic matter. If your warp drive requires laying down tracks in advance then you can only go back to when a landing pad was placed. But even then, if you make a landing pad and then start making tracks to connect it to something else you don't know if the thing that came out is from your own future. This is not physics that has been tested yet. In fact it is the opposite, it involves just making whatever warp drive, wormhole, or time machine you want, And then asking how weird (or exotic) your matter needs to be to allow it.
Since you started by writing down whatever you want the only limit is the laws of physics. And in general relativity, the physics applies just to little blobs. In general relativity there are solutions with time travel that have zero matter in them at all (exotic or regular), but there are other solutions where the two of them agree for a trillion years but in one of them spacetime is boring and never ever develops time travel. So you could get time travel if the universe just looks the same. If it looks exactly the same, it could repeat what it did last time, or it could literally be the past. They are different solutions, but they make the same predictions. When you have exotic matter you can try to force connections between regions that otherwise would not be connected. For instance Earth now and Earth in five years with a stop at Alpha Centauri in between is not a normally possible path without exotic matter. But if this kind of technology leads to arrangements where paths can go from here to the future and back to here that is a problem. And all these paths are in the 4d spacetime, so there are solutions like that. But it isn't just bad because of time travel, some people are concerned that as you start to assemble devices close to the arrangement that allows time travel then they could break, and since they worked by exotic matter, they break by and explosion that has regular energy from the explosion overwhelm the exotic matter.
Exotic matter is exotic for having negative energy density. So if to stop it you need to flood the region with positive energy, so if there is energy accessible from around it and that is then forced to explode so it can flow to the negative energy regions (where the exotic matter is) this could be bad even before any actual time travel starts, and maybe there is no time travel, just explosions.
And a similar thing happens for a warp drive. Some people think the front end builds up some positive energy which can then take off when you slow down or stop and that it shoots lots of energy off in the forward direction.
Now you could use a warp drive to go from Earth to Alpha Centauri and back in 8.8 years (just under lightspeed) and no time travel is possible. You still have lots of debris between here and there and if you pile it in front of you and then release it when you slow down at the end you will still hurt whatever is at your destination. And you still need exotic matter?
Why? You could make that trip with rockets so why do you need exotic matter? Because that solution started out assuming you aged 8.8 years. And if you used a rocket you would age less than 8.8 years and the people staying home at earth would age 8.8 years.
But that is the while point. You'd have to say when-where you start, you'd have to say when-where you stop, and which when-wheres in between you want to stop at along the way.
If you then also say how much you want to age on your trip then if that is what you could age with rockets then you can do that. Otherwise our only option is to literally change the regions of 4d spacetime in between the when-wheres you wanted to be to make you age the rate you want or to connect those regions at all if they weren't normally connected.
And as always these are 4d regions. So this is a when-where we are talking about being connected to another when-where.
What happens to the warped space?
The region with the exotic matter can't have too much regular matter in it or it won't be exotic anymore (it needs to be on average a certain amount exotic) so your options to are to lay your tracks around the stars in the space in between like a subway that has to avoid some things. And then you either divert thibgs around you or pike them up in front, and the latter might be unavoidable if even when you make an actual sealed tunnel the vacuum on front of you unavoidably has things in it. But that is quantum gravity, so we don't really know how that works.
Would the entities like stars, gas clouds and planets present in the warp be squished together or thrown apart because of the warp.
The warp drive solution was designed to only affect the thibgs right in the path. So imagine a tube and only the things inside the tube are affected.
Wouldnt it make the warp drive more dangerous to the Universe and would the Universe prevent it?
More dangerous than time travel? Its the other way around, having regular matter enter your warp drive would make it not exotic and make it fail. So don't build your train tracks to go through any stars.