Here are a few random notes about rules to do with genders.

When writing to a mixed group, you can use a slash mark to alternate genders (similar to our “If your student wishes for a bought lunch, he/she can…”).
Example: Mig langar til að biðja þig að vera svo góð/ur að svara eftirfarandi könnun.
Translation: I want to ask you to be so good as to answer the following survey/questionnaire. (“Please answer my survey”)
You can see that they wrote þig, which is “second person singular” (“you”) and not þið, which is second person plural (“you all”, “you guys”). See below.
Another example: Komdu sæl /sæll ágæti þátttakandi.
Translation: Hello dear (lit. “fine/good”) participant.

I’m not sure I’ve ever seen þið form used by Icelanders when addressing a general online audience such as an announcement or welcome page. Þið seems to just be used when talking to specific people, such as in “you two people who posted on this forum thread before me” or when they’re calling people out specifically by their (user)names. In my experience, I’d have to say that if someone uses it then it’s probably someone posting online to specifically talk to a few people (say, a grandma with a blog that only her daughters read) instead of to a general audience, but I’m no expert. It might be that you use it the same as in English, where “you” is for a single person or very large/unknown amount of people (such as a television host addressing the entire nation), and “you all/guys” is for a smaller group of people but still at least more then one person.

If you don’t want to use slashes you can use masculine form just like in English. “If your student wishes for a bought lunch, he can…” would be the same thing, you would just automatically know that it’s “he or she”.

Example: Ef nemandi er veikur ber hann sjálfur ábyrgð á að verða sér úti um upplýsingar og að æfa það verk sem hann missti af.
Translation: If a student is sick, he himself bears the responsibility of getting a hold of the information and exercises from the work which he has missed.

In some cases you can make compound words to make something clearer, even if this isn’t often done:
barn – child (can be a male or female child)
stúlka – girl
stúlkubarn – “girl-child” although hopefully without the negative connotations “girl-child” has in English.

The names of letters are neuter gender, except for z (seta) which is feminine:
Examples: Stórt A (capital A)
litið a (lowercase a)
essið er stafur (The S is a letter). “Ess” is the spelled-out name of s in Icelandic, notice how “the”, in this case the -ið on essið, is in neuter form.
setan er notuð í ensku (the z is used in English). “The”, which in this case is the -n on setan, is in feminine form.

When completely using a foreign word, such as when writing or saying a foreigner’s name, often it will just be un-declined/un-conjugated or treated as neuter gender. This is especially true if it’s uncommon or doesn’t end in something that’s a real Icelandic name-ending (such as -ur). The fact is that early on people used to Icelandify/foreignize names a lot more often. Icelanders did it to foreign names in general (such as in translations), foreigners visiting Iceland did it to their own names (such as merchants), and Icelanders used to foreign-ize their own names when abroad (such as students studying in mainland Europe). The further back you go the more often they did this, but today this is almost exclusively done when translating children’s books and sometimes as people’s nicknames for their foreign friends, as far as I can tell.

The rules for deciding which gender to put them in, how to Icelandify them, and how to decline them are basically made-up based on the historical evidence of who did what earlier on, or how they Icelandify words nowadays. A lot of those guys in the past just did things on whims anyway. For example one old book I have actually switched translators halfway through and so for the first half they Icelandified and declined most of the foreign names, and in the second half they either Icelandified them differently and didn’t decline them, or didn’t change them at all. Thus even if you can say there are loose rules based on historical trends, the overall majority of things on the matter say you can basically do as you see fit.