Is the grammar completely correct as well? If so, I'm surprised I got it right!

No.
Not 'm'argent' but mon argent (singular nouns beginning with vowels always take this form) but better still 'ma vie'
not 'avec' but 'de', but better still 'en + a present participle'
not 'sites Web' but 'sites d'internet'.
Therefore:
je gagne ma vie en créant des sites d'internet.

I'm not a native speaker so you can probably disregard me but "En ce moment" sounds a little formal or stilted to me.

Len is right - by saying "en ce moment" you convey the idea that this is temporary; you are implying that in the future you will be doing something else. By the way, even in English, the phrase has an odd ring - is web design and building a career; a part time gig.

Also, "conception" and "création" are not the same thing. The first is more high-brow (design). The second is more bricks and mortar (build).

Well I earn money with it, but it actually is more of a part-time gig. A hobby gone wild.
I also live with my parents and they pay for lots of stuff, so it's not like I'm 'making my living' that way, because my parents make my living at the moment.

Holding all of that in mind, how is:
En ce moment, je gagne mon argent en la conception des sites d'internet.

Thanks so far?

Minus question mark.

I'd still personally not use "conception" even if you are doing design. So "en créant des sites".

Also "À l'instant" sounds better to me than "en ce moment" but you may prefer to wait for a more proficient speaker to confirm. Also I can't help but feel that "je gagne mon argent" sounds super anglo, but again...

Also, question for thomas (or anyone else): I've always heard "sites-web" rather than "sites d'internet". Is that not courant?

No.
Not 'm'argent' but mon argent (singular nouns beginning with vowels always take this form) but better still 'ma vie'
not 'avec' but 'de', but better still 'en + a present participle'
not 'sites Web' but 'sites d'internet'.
Therefore:
je gagne ma vie en créant des sites d'internet.

Sorry. Posted twice by accident.

Yes. "Je gagne ma vie" gives more of a sense of a full-time profession. If it's "a gig", substitute "je gagne de l'argent"

En ce moment, je gagne de l'argent en la conception des sites d'internet.
En ce moment, je gagne de l'argent en créant des sites d'internet.
En ce moment, je gagne de l'argent en la conception des sites web.
En ce moment, je gagne de l'argent en créant des sites web.

One of those? Something else?

French is my native language.

• "En ce moment" is perfect and not stilted at all. "À l'instant" would not be good because it means "At this very moment".

• "je gagne de l'argent" is perfect and is better than "je gagne ma vie" which would be used for a 'permanent' job.

• "en créant des sites" : in this sentence, you need a gerund, which is formed with "en" + present participle. Also, in this case, you should use the verb "créer" even if you are doing design.

• "internet" without the preposition " d' ". "web" would be acceptable but much less than "internet".

→ "En ce moment, je gagne de l'argent en créant des sites internet."

Well, I think I'll just believe the native French guy! Thanks!

Not to contradict the native speaker, but all I've ever heard from native speakers was "site web". Eg "twitter est un site web". Otherwise I agree. Maybe it's more of an African thing. I have found that non-French native French speakers are more willing to use anglicisms, especially in technology, than the metropolitan French are - perhaps due to the influence of the Academy. For example "l'homme d'affaires" vs "le businessman"

Another one: Do you love/like sports?
Aimez-vous des sports?
Correct?

Formally:
Aimez-vous les sports?
More colloquially:
Vous aimez les sports?

The word "sport" should be used in the singular, even though it refers to sports.

→ "Aimez-vous le sport ?"

Also, please note that, in French, signs of punctuation made up of two elements which are on a vertical line should have a space before (this is not the case in many languages) and after them. These signs are :

: ; ? !

Dis: "tu aimes le sport?"

To be formal: "est-ce que vous aimez le sport?"