If I get the web-design done by someone else, how can I ensure he is not simply passing me some recycled work or blatantly copying/pasting somebody's (or his own) work? There really is no way to make sure, right? So how can I ask a web-designer from one of those freelancing websites to get the work done? How does this entire process work? 
How to ensure a web designer is selling you an original design
website design
If you hire a real web designer rather than crowdsourcing it, then you should be able to see the design process and be involved in it.Usually, the designer will send you some mockups and update you with their progress (or you can ask for updates) and ask for feedback on the current design.There's no guarantee that the designer isn't stealing from the work of other designers (though few designs are truly unique or entirely original), but if you're having a custom-tailored website designed, then there shouldn't be much risk of this.A good designer will not go out of the way to steal a template just so they can modify it beyond recognition. It would be easier to just use the existing designs or templates as creative inspiration and build an original design from the ground up to the client's needs. It's just not fun to rehash other people's work.A poor designer, on the other hand, won't be able to emulate the style or design quality of the original designer of a template they're passing off on their own. So the second you ask them to make major changes to the design, they'll give themselves away. A poor designer will make a good design look worse and worse the more they make changes to it.