If we're talking about fiction, acting irresponsible and stupid does not disqualify one from being a hero, nor does it make one a villain. Being incredibly monstrous or neglectful to your own family does not automatically make a character a villain either. Just consider the story of the Greek hero Heracles. As we all know, he killed his whole family. He spends the rest of the story doing crazy tasks for a selfish king as penance. Heracles was considered a hero by the Greeks and he's still considered a hero now. Where parenting is concerned, Goku seems pretty harmless by comparison. He may have gotten his sons and wife killed, but he never killed them himself (not in the current timeline anyway, though the whole Zamasu thing is actually quite analogous to Hera sending madness down on Heracles).
Whether a character is hero or villain all depends on the framing of the story, as well as the heroic traits that are valued in the culture the story comes from.
Oftentimes in fiction, the terms hero and villain are synonymous with protagonist and antagonist (not always, but often). In Dragon Ball, Dragon Ball Z, and Dragon Ball Super, I think it's pretty clear Goku is the hero and our protagonist. The story does not portray him as a villain - just a careless idiot who sometimes gets carried away wanting to fight strong opponents.
As to addressing your specific matters in your bulleted list:
- Goku is a warrior and he likes fighting. These are not inherently villainous traits. Look at all the heroic warrior stories we have: Beowulf, Heracles, Achilles, etc. Achilles's selfishness gets his friend killed, but he's still considered a hero in spite of having this negative trait.
- Mercy and forgiveness are not considered a villainous trait. They are considered heroic traits. The consequences of the forgiveness don't really matter for determining a hero. Batman is still considered a hero even though he keeps showing Joker mercy, which results in more people dying.
- Giving Cell a Senzu Bean was idiotic and irresponsible. Those traits don't make Goku a villain. 6. and 7. are just Goku being stupid and/or irreverent too. Doesn't make him a villain.
- I have more to say about the Tournament of Power below.
- "He didn't care about his own family." This is not true. You could say he is neglectful, though there are some who would contend this too. I won't, though. It doesn't matter whether he's a good father and husband or not, as I've already explained.
Let's look at some definitions of hero and villain from Merriam Webster.
a mythological or legendary figure often of divine descent
endowed with great strength or ability
an illustrious warrior
a person admired for achievements and noble qualities
one who shows great courage.
the principal character in a literary or
Nowhere in there does it say Goku has to take only good actions, or that he is not allowed to take evil actions. And note that with how ambiguous the world is, it would take a philosopher to figure out what is good and evil anyway. For instance, is it really evil to give Frieza a second chance? We have the hindsight of knowing he won't reform, but Goku doesn't.
1 : a character in a story or play who opposes the hero
2 : a deliberate scoundrel or criminal
3 : one blamed for a particular evil
4 : villein
5 : an uncouth person : boor
These ones are worth going through, which is why I've kept the numbers in. Goku definitely is not 1, because he is the hero of the story. He is not 2 either, because none of the evil he does is deliberate. But for 3-5, there is actually a case for each one that Goku is a villain.
- Many of the other universes blame Goku for the Tournament of Power.
- We've seen Goku do some farming in Dragon Ball Super, so go ahead and call him a villain. He most likely does not actually have a lord though. Or maybe the land he lives on actually belongs to the Ox King.
- Goku has been shown to have poor manners at times.
Is Goku a villain? It depends on what you mean. There are some who considered him a villain for a time, namely those from the other universes forced to participate in the Tournament of Power. As you say you believe Goku is "the worst", however, I do not think this is what you mean by villain, in which case, I think we have to conclude that he is not one. We can reasonably conclude he is a hero though, as he fits most of the definitions provided.
If the question is whether Goku is a boorish peasant though, then the answer is yeah, kind of. Though he probably doesn't fit the literal definition of a peasant, he fits the bill enough that it makes sense to use that as an insult against him.