Do all stars have an Oort cloud like ours that will be filled with comets and other objects? If not, why are they not around every star?
Awesome question, especially since we know so little of the answer.
Nobody knows for sure how the Oort Cloud formed - I'll put that out there right now - but the current hypothesis is that it was originally part of the Sun's protoplanetary disk. All of the ice and rock coalesced into small bodies - proto-comets, if you will. While these bodies were much closer in to the Sun than they are today, they were tossed far out by gravitational interactions with the gas giants. Other interstellar comets could also have been captured by the Sun, adding to the population.
So why is the Oort Cloud spherical? After all, the protoplanetary disk was just a flat disk. Why were the orbits of the objects perturbed? Well, the Oort Cloud objects are only loosely bound to the Sun - relatively, that is. They can be influenced by passing stars or other objects. It appears that galactic-scale tidal forces, combined with the influence of passing stars, molded the Cloud into its current spherical shape.
So what does this all tell us? Well, we know other stars have protoplanetary disks, right? Some also have exoplanets - gas giants like Jupiter. They are also subject to tidal forces and the passing of nearby stars. So, theoretically, there's no reason why other stars shouldn't have Oort Clouds.
So can we find them? The answer is, most likely, no. Here's why. According to Wikipedia,
The outer Oort cloud may have trillions of objects larger than 1 km (0.62 mi), and billions with solar system absolute magnitudes brighter than 11
An absolute magnitude of a solar system object of 11 is very dim. Now, the object's apparent magnitude is how it would look from a given distance; the absolute magnitude is how it looks from a distance of 1 AU (in the case of Solar System objects, this quantity is denoted $H$). Oort-cloud objects are 2,000 - 50,000 (or more) AU away-- so these objects, to us in the same solar system, have an apparent magnitude much fainter than 11.
The point of that poorly-explained interlude is that these objects are faint. Very faint. And objects in Oort Clouds around other stars would appear even fainter. Using the distance modulus, we can calculate the apparent magnitude of an object if the distance to that object and its absolute magnitude are known:
where $m$ is apparent magnitude, $M$ is a scaling of $H$ normally used for stars, and $d$ is the distance in AU.
Given an Oort Cloud object $x$ light-years away, you can figure out how bright (or dim) it would appear, given that 1 light-year is 63241 AU. Try this with the distances of nearby stars, and you'll realize how dim objects in these stars' Oort Clouds would be.
As a final note: We don't know for sure if other Oort Clouds exist. From what I've been able to find, we don't have sufficiently powerful telescopes to observe these hypothetical Clouds, and so we don't (and may never) know if they exist.
I hope this helps.
As I found from a link from an answer to this question on Physics, we've found Kuiper-Belt-like disks around other stars. This means it is certainly plausible for these stars to have Oort Clouds, too. And exocomets have been detected, which is another good sign.
All stars could have their own Oort cloud, but all stars don't. As HDE says the Oort cloud was formed by material in the sun's protoplanetary disk and interstellar comets that were caught by the sun. Some theories say that almost all comets were formed around the Sun, and this wouldn't allow us to say much about comets around other stars. However there are others such as Levison et al. in Capture of the Sun's Oort Cloud from Stars in Its Birth Cluster that contend that a majority of the comets have to have their origin from other stars' protoplanetary disks. This must be so because the current models fail to explain the amount of comets in the Oort cloud.
The sun was born in a star cluster in close proximity to other stars. These stars would have to be the source of any significant contribution to the Oort cloud by stars other than the Sun. That is why Levison et al. tested their hypothesis by constructing a young star cluster and simulating its dynamics with an n-body simulator. They found that their hypothesis was confirmed by the simulation. If every star in the star cluster starts with some comets in their respective protoplanetary disks, then some stars collect many more comets from their neighbors before they leave the star cluster, as the Sun did long ago.
I recently wrote an N-body simulator to replicate their paper. The result is not available in English, but the experiment is described briefly here. There is an animation there that will help you understand what I am getting at. My simulator ended up being a lot slower than theirs so I couldn't have nearly the same number of objects in my simulations, but I observed the same tendencies as they did.
To answer the question, I did see that some stars lost all their comets and I also saw some stars that gained comets. It all depended on the dynamics in the star cluster. Some left it very early with just the comets that formed in their own protoplanetary disks, some left it with more comets than that and some left it with no comets at all. Based on these simulations I would say that, as I started by saying, all stars could have their own Oort cloud, but all stars don't.