Into should also be accepted as the preposition - unless there is some hidden nuance in the Polish phrase that I do not understand.
No. That is "witać".
- Witamy w naszym domu! (Welcome to our house!)
We sometimes use the form of the 2nd person future tense. It's a bit like "we welcome" changing to "you are welcome"? That's not exactly that, but similar. Anyway, instead of "witamy" we can say "witajcie", with 2nd person plural future.
I have found in these exercises there is a grey line between the use of verb + dative as opposed to verb + accusative. Given the invitation is for someone, I could have easily expected the use of "wam" as opposed to "was". Are there any hard and fast rules for determining whether dative is used as opposed to accusative?
Well, "to invite" just takes a direct object. Compare "I invite you" vs "I am sending an invitation to you" or "I am sending you an invitation". sending an invitation takes a direct object (invitation) and an indirect object (the recipient).