Is it allowed and safe to carry a passenger / non-pilot in the front seat of a small general aviation airplane?
There should be no problem in general with flying with a passenger in a front row seat in a GA flight, as long as the aircraft is certified for single-pilot operations. You'd want to check your national regulations to be certain, but I have a hard time imagining that it would be prohibited as long as you're allowed to carry passengers in the first place. In general, it's normal practice in small airplanes for passengers to fly up front.

However, especially for anyone not used to a cockpit environment, you would definitely want to brief them on expected behavior before beginning the flight, in addition to any pre-flight briefing required by legislation. Even if such legislation only require briefing before a commercial flight, it might be good, and put your passenger more at ease, to do something similar before a non-commercial flight as well. (How to get out of the seat belt, how to exit the aircraft, where the airsickness bag is, ...)

None of this needs to be complex stuff; what I have in mind here is mostly things like keep your feet clear of the pedals, keep your legs clear of the yoke, don't touch any of the controls unless told to, and similar. As Jamiec points out in their answer, it's very unlikely that someone not used to light aircraft would know what the rows of instruments, switches and controls are about, so a quick "fly safely in the cockpit" style briefing would absolutely be appropriate in addition to a general pre-flight briefing.

For a (somewhat humorous) example of what I have in mind, you might want to check out Preflight Announcement, Maule on Youtube.

Especially since this is someone you know, and you'd therefore presumably be able to gauge their reactions more easily than if they were a stranger, don't be afraid to sprinkle some appropriate humor into the briefing. For one example, I've heard of the airsickness bag being referred to as the rollercoaster bag.

It may also help put your passenger at ease if you specifically tell them what you're about to do before you do something unusual or non-obvious, such as the engine runup and magneto check. Particularly with non-pilot passengers, you don't want to get technical; something like "I'll do a few engine tests to make sure it's running smoothly before we take off" is more meaningful to them than "I'll bring the engine to 2391 RPM while holding position, and then check the magnetos and carb heat before reporting ready to Tower". Certainly when flying with another pilot you might just say "mag check", or even nothing at all (they'd probably be expecting you to do it).

Especially if your friend is an adult or an able teenager (but probably not for people much younger than that), do consider the possibility of enlisting their help with simple tasks during the flight. Regulations permitting and of course assuming that you as the pilot in command feel comfortable with it, you could even let your friend take the controls for a while, as long as you're able to take over if necessary and you're able to ensure that nothing untoward happens. If nothing else, they can definitely help look out for and call out nearby traffic while hopefully enjoying the view.

You could also show your friend how to use the radio to contact ATC in an emergency, and at least the very basics of keeping the airplane in the air (things like mostly straight and level flight; shallow turns, possibly without rudder; climb and descent; stall indication and response; where the throttle is and how to operate it; and possibly the IDENT function on the transponder). In the unlikely event that something were to happen to you, while not a guarantee, doing so would greatly increase the odds of both of you eventually making it out of the airplane in one piece. I have written a much more in-depth answer about this from the passenger's perspective which you might find informative. It's possible that there are even courses specifically tailored toward GA passengers in your locale, which would/could cover such topics; if not, any instructor worth their salt should be able to put something together easily enough for a given airplane type. If this is someone you expect to fly with often, that might be money well spent for both of you.