Any form of "swój" refers back to the subject of the sentence. The subject of this sentence is "We": We help. So if we help "swojemu" dziadkowi, that means we help our grandpa. If "Oni pomagają swojemu dziadkowi", then they help their grandpa. And so on, and so on.
"Pomagamy naszemu dziadkowi" is perfectly fine.
Basically, if you can use a form of "swój" correctly, do it. Some 'normal' forms (mój/twój/jego/wasz etc.) feel more natural than others in a sentence which could use 'swój'. So 3rd person possessives actually suggest that this "his/her/their" is different than the subject of the sentence. That Adam is cleaning John's car, not his own. 2nd person possessives used instead of "swój" seem very clumsy. But 1st person are actually pretty okay.
I am not (yet) very proficient at Polish, but I have studied German grammar at a reasonably high level at the university. I think this way of conceptualizing dative verbs may be confusing to learners. As far as I know all indirect objects require the dative in both German and Polish. This is the case after verbs like dawac/geben or opowiadac/erzählen, where the direct object is in the accusative and and the indirect object is in the dative. However, quite a few verbs require that the DIRECT object have the dative form. Interestingly, the most common verbs that this applies to are the same in both Polish and German: congratulate, help, fit, answer, thank and quite a few others. I may be mistaken here, but I don't think there are many (any?) schools of grammar that would say that the dative in this sentence is caused by the noun being the indirect object in this sentence. It is a direct object in the dative because the verb governs the dative.
I just read that "To have an indirect object in a sentence there must first be a direct object.", so I guess my definition is too simplified. In Polish there word "object" in terms of grammar translates to "dopełnienie", and there are two types: dopełnienie dalsze ('farther'?) and dopełnienie bliższe ('closer'). But for both of them there are several cases they could be used.
If calling it an indirect object is wrong, then so far I don't have a better explanation than saying that "pomagać" takes Dative. But it's interesting that no one corrected me so far...
Strangely enough, the email notification shows me a different reply from you, maybe you have changed for the current one. Anyways, I am fine with this rule; sometimes the rule is just "it is this way because the language is like this" and several languages have such rules (portuguese is riddled with them).