If you do a little reading about fume cupboards, and in particular, benchtop extraction hoods, you will discover that they are constructed from materials not too dissimilar to your design; consider the difference being amateur use versus professional use. To address your four main concerns:
Is the thick plastic sheet enough the prevent most fumes from diffusing in my workplace?
Provided the plastic is thick enough. So, certainly thicker than cling wrap, and I'd be happy using the types of material common to decent portable greenhouses. It needs to be robust enough to withstand the extraction forces from your exhaust fan. If your exhaust is working correctly, any chemicals will not be in contact with the plastic long enough to be a concern. Be sure you run a post-purge extraction once you have finished your day's chemistry to remove all traces of chemical fumes.
Is this plastic sheet a potential danger in case of fire? Is there some fire resistant (at least until I'm able to pull my fire blanket) plastic sheet that should (must) be used?
Most fume hoods are made from combustible material. I have, in my 20+ year career as a research chemist, seen many fume hoods destroyed by fire. This includes hoods made from wood, various forms of plastic, even those with metal frames and plastic components. If you have a fire, yes the plastic sheet will burn, and depending on the material, it may burn quickly or slowly - you will have to judge that. This, then, will increase the volume of your fire, and the amount of combustible material. So you will need to manage that in your workshop layout. So, yes, this is a hazard, but one that you should be able to plan for, and adequately implement a risk mitigation strategy. But the real concerns are not whether the plastic sheet would be a danger if there is fire, but (a)will the plastic sheet increase the risk of fire, and (b)what effect will your exhaust fan have in the event of fire. If you are doing chemistry that involves refluxing, as your picture suggests, you need to make sure that the size of the hood is big enough that you are not heating your plastic, and that the air flow is sufficient to extract any hot air from the roof of your hood. You also need to consider that fires in fume hoods are generally more active than fires on open bench areas because they are actively fanned and have a fresh air source driven into the fire zone because of your extraction fan. A fire blanket in your workshop is not a sufficient measure against the type of fire you might have doing chemistry, Buy yourself at least one dry chemical extinguisher, which are sold in all major retailers that sell fire blankets. If you are using chlorine, then you should NOT use the ABC type dry extinguishers that use ammonium based material, as they will react violently with chlorine. You should use a BC type, which is based on sodium bicarbonate.
Should I be worried about damaging the environment with these exhaust fumes as there is no filtration in this system? Or is reading the MSDS (i.e. the environment section) for the chemicals being used enough?
Yes, you should be worried. Reading the MSDS is not enough. You need to read it, understand it, and then take appropriate measures to mitigate any hazardous risk to the local environment. If you are extracting chlorine gas, where is it going? Apart from being pulled through your pump system, which will be corroded pretty badly after only a few uses, what is directly outside at your pump exhaust? If you position your exhaust just outside a window of your building, what local wind channelling exists to push that exhaust straight into the next window of your, or your neighbour's house? You could consider installing a carbon filter like the types used on water systems, as these are not overly expensive, but for many commercial systems, no scrubbing or filtration occurs, but the key issue is where is your exhaust going? So, understand your design, consider your risks, and plan your exhaust appropriately.
There is said nothing about the necessity of an explosion proof fan in the build instructions. To what extend should I worry about this when sucking off flammables?
I'm pretty sure that all building codes (certainly all the ones I have ever encountered) will specify that any motor used in an area containing flammable vapours (and dust, if you need to consider that for your woodworking) should be of the type approved for such an environment. What dust extraction do you use for your woodworking? If you get an arc from the pump motor, you will very likely get ignition. Again, this is something I have seen in research laboratories for all sorts of non-approved electrical equipment, including pumps, air heaters, fridges. So, if you are exhausting flammable vapours, this is a real risk.
The final consideration you haven't really mentioned is the exhaust pump itself. You need to make sure that it is of suitable size to provide correct laminar flow into the front of the hood, around all your equipment, into the exhaust outlet, and pushed back through the exhaust port after the pump. You should test this using a smoke pellet, or something similar. the larger the opening of your hood, the larger the exhaust fan needs to be, which is why you should always lower the sash hood on fume cupboards when doing chemistry.