Then that seems to be a Brazilian thing as all the Portuguese people I have talked to about você and the 2nd vs 3rd Person have all insisted it is a 3rd Person treatment pronoun (which they actually hardly ever use because they consider it a bit crude) and even expressed a bit of shock that anyone would consider it otherwise.
Just my opinion. Others experiences may differ.
"Dad" is a way to refer to one's father in an informal way (not that cute). Adults refer to their father as "Dad".
"Daddy" is the way that children refer to their father. (very cute)
"Papa" or "pop" sounds old fashioned to me, but perhaps some people from the southern states in the USA still say that. I'm not sure.
A transplant to Texas here, but as a former Army brat I traveled all over the US so I have a few observations.
"Daddy" is common among young children, whereas "Dad" is most common with older children and adults.
My father always called his dad "Pop" and that seems more common in the rural Pacific Northwest, where he is from. I've heard "Pop" in other rural places too. It might be more of a city vs small town thing.
I don't know the South extensively, but in Texas most older children call their father, "Dad". My younger children refer to their father as "Papa", while the teens say "Dad". I think that is due to the family French influence though.
This is amusing. "Hipster" dads want to be called "papa".
My daughters call their father "papa" and he is decidedly not hipster...just European.
(1) Yes, you can leave "eu" out of the sentence. Due to its ending, "amo" can only be related to "eu", but when a verb may be related to more than one pronoun/subject, you should add the pronoun to avoid ambiguity.
(2) Portuguese, Italian, Spanish and French are pretty much similar =)
(3) "Ti" is used after prepositions, as in "Tenho grande admiração por ti". "Ti amo" is not grammatically correct, but it is allowed for poetry, song, etc. It's what we call "licença poética" (http://www.infoescola.com/literatura/licenca-poetica/ -- http://www.recantodasletras.com.br/teorialiteraria/1244135). Also, "te" is pronounced like "ti". =)
Lhe! A quotation:
"...mencionemos a tendência atual dos brasileiros em transformar O/A em LHE. Essa invasão de área está acontecendo porque: 1º) o O tem pouco corpo fonético, baixa audibilidade, e portanto comunica mal; 2º) no conjunto dos pronomes oblíquos, LHE se encaixa melhor do que O: me, te, se, lhe, com a vogal (e) apoiada por consoante, em vez de uma só vogal (o, a), formam uma sequência mais espontânea."
Te amo is a formal way to say "love" in Spanish.... typically used with family and husband/wives... Are you Mexican, because I feel pretty confident that is the same everywhere (my ex is venezuelan and I live in Spain and its the same). Querer can be used more for friends and bf/gf (before it gets too serious).