What you propose is potentially possible (with lots of caveats!). It wouldn't physically damage your computer but quite possibly there would be software conflicts resulting in problems. Since windows 7 has only recently become unsupported it's probable (though nothing is ever certain) that your forthcoming machine would run on it. However, as trader 333 has suggested – it would be useful to know the full spec of your machine. I've just seen that you are using a Ryzen 5 2600 X which is not the latest version of the CPU so I don't see why it shouldn't work with windows 7 - but that's only a guess, you need to check.
I'm not aware of what a "trial version of Windows 10" is. Is it possible that the machine is being supplied with an unregistered/unactivated copy of Windows 10? I would expect any new machine supplied professionally to have a fully working & activated copy of Windows 10 on it. You can download (free) legitimately from Microsoft a fully working and up-to-date copy of Windows 10 and install it on any suitable machine. You will be prompted to activate it (either with a legitimate code or by paying them some dosh) but it will still continue working satisfactorily while unactivated, indefinitely. Microsoft seem to have a different policy now whereby they do not cripple your machine if you do not have an activated copy of Windows. If you continue to run it unactivated the only disadvantage is that you will have a very discreet reminder in a corner of your windows screen and you will be unable to personalise the colours of your screen et cetera. If that bugs you, you can pay up!
Although the upgrade route from previous Windows versions has "officially" been discontinued, the fact is that it is still there and works ok – I've upgraded several machines since the deadline passed. The upgrade route presumably is what you are alluding to and there should not be a problem in principle in upgrading from windows 7. However, ............ the official upgrade route will take into account the type of licence that the installed version of Windows 7 has. This can get quite complicated and basically, if your edition of Windows 7 was a full retail copy then you should be okay but if it was originally supplied with a machine "as is" it's probably what is called an "OEM" version which would have been "married" to the original machine upon which it was installed and probably will not register/activate to another, unless that's the actual machine being upgraded. The only real way to find out with upgrades to Windows 10 is to actually try it – sometimes it will work when it shouldn't & vice versa. If you've actually paid for a legit copy of windows 7 then the Microsoft helpline are usually very helpful and you shouldn't have any problems – and sometimes you only have to go through an automated telephone procedure with no human involved.
If your machine is arriving with a load of gaming software et cetera you need to be sure that you can reinstall it if the upgrade goes wrong. If you are not completely happy with the process I would suggest that you find someone who is able to help you. But better than that, I would get the supplier of your new PC to get it all sorted out for you upfront.
