Thorn comes from the runic alphabet; yogh does not; it derives from a particular way of writing the letter g in manuscripts from the British isles which looked very much like 'z'; so it is part of the Latin alphabet but not Latin language-- (it accounts for the apparent g-z confusion in mingus/menzies etc.)

Thorn was in use in Old English manuscripts before anyone started to write Icelandic prose on parchment--but derivation from Greek theta is a conjecture--thorn derives from a rune in the older runic alphabet, which would have been used by those who used runes... long before the English or the Scandinavians has learnt latin.

It's hard to say that Iceland had a 'Roman' influence on its spelling. When Icelandic began to be written with the Roman alphabet, latin was no longer a spoken language, but those who began to write it in with the latin alphabet instead of runes had themselves learned latin--their own mother tongues would have been English, German, even Irish.
Ultimately, if you accept the idea that the alphabet was invented once, then it is likely that the runic alphabet developed through contact between speakers of a (proto) germanic vernacular and people with latin literacy.