The verse John 3:16 makes use of two grammatical topics which are important in both Greek and Latin: a result clause and a purpose clause. According to this, the verse can be logically divided in two. I will first treat your handling of the result clause ('For God so loved the world, that he gave his only son') and then, if I find time, I will edit this answer and add a part about the purpose clause ('so that all who believe in him shall not perish but have eternal life'). Furthermore, I will reproduce the Greek and use that to verify my claim. (Reproducing the Vulgate might only spoil the attempt.)
οὕτως γὰρ ἠγάπησεν ὁ θεὸς τὸν κόσμον, ὥστε τὸν υἱὸν τὸν μονογενῆ ἔδωκεν, ἵνα πᾶς ὁ πιστεύων εἰς αὐτὸν μὴ ἀπόληται ἀλλ’ ἔχῃ ζωὴν αἰώνιον. (Nestle-Aland John 3:16)
The first part (οὕτως γὰρ...ονογενῆ ἔδωκεν) is called a result clause. The Greek word οὕτως ('so') is a demonstrative which often introduces a result clause, and ὥστε ('that') is a conjunction that often follows.
Result clauses also exist in Latin, but you form them differently. Instead of using 'quod', as you do in your translation, you should be using 'ut'. The verb following 'ut' should be placed in the subjunctive. You can read more about Latin result clauses online, such as here.
Within this result clause, there are a few details I would like to address. These details concern your translation of the English words 'For', 'so', and 'that'.
Regarding 'For' (γὰρ), I suspected that 'Quia' was not best. I consulted the Vulgate, and found that it uses 'enim' instead. This is probably the best translation of the Greek particle γὰρ. It makes sense to translate a particle with a particle, so you might want to research the subject of postpositive particles. 'Enim' is a frequent one in Latin, especially the Vulgate. So is 'autem'.
A postpositive particle is normally placed after the first word of a sentence, so it becomes the second word. It would benefit you to consider briefly how these particles can give a sense of punctuation, in a language where there often was no punctuation. When you reach an 'enim' or an 'autem', then it's reasonable to guess that it's the second word in a new sentence.
You might also want a Latin word which captures the demonstrative adverb, 'so'. ('For god so loved the world.') I consulted the Vulgate and 'sic' is a good choice here. You could also use 'tam' or 'ita'. These words correspond to the Greek adverb οὕτως. You can verify this information about demonstratives in result clauses in the same source I linked to earlier.
Thus, for the first part, we get:
Sic enim Deus mundum amabat ut filium unum et solum offerret...
The word 'offerret' is the imperfect subjunctive, since, if you remember, we have to use the subjunctive mood in the 'ut' part of the result clause. However, I think a better verb might substitute for 'offero'. I checked the Vulgate and it uses what I suspect: dare. I will leave it to you to translate this verb (do, dare) to the correct tense and mood, if you choose.
That covers the first part of the verse: the result clause. The second part is the purpose clause, which requires a second use of the conjunction 'ut'. I will post the first part of my answer for now, and if I have time, I may edit this answer and add the second part, about the purpose clause. Hope this helps.