Imagine someone make a neder not to eat pork (this is not a general Neder not to eat meat rather specifically prohibiting pork which is already forbidden). Is that neder valid?
If he then eats pork, is he also responsible for not respecting his neder or only for transgressing Halakha of not eating pig?
The Gemara in Sh'vuos 22b:
שבועה שלא אוכל, ואכל נבילות וטריפות, שקצים ורמשים - חייב, ור' שמעון פוטר; והוינן בה, אמאי חייב? מושבע ועומד מהר סיני הוא! רב ושמואל ור' יוחנן דאמרי: בכולל דברים המותרין עם דברים האסורין
Someone swears not to eat, and they eat forbidden foods, they are responsible [for violating their oath]. Why are they responsible for their oath? They are already responsible for this from Har Sinai! Rav Shmuel and R' Yochanan said: because he included permissible things with the forbidden things.
An oath that only includes things that are already forbidden is not valid, because he is already "sworn" into not doing these things. There are various exceptions, such as if his oath takes effect because it includes things that are not already forbidden from Har Sinai.
Answer to question one:
A person Can make a neder to make the pork ossur (prohibited) to eat for himself, and it works and is not a superfluous neder Nedarim 8a:
מנין שנשבעין לקיים את המצוה שנאמר נשבעתי ואקיימה לשמור משפטי צדקך והלא מושבע ועומד מהר םיני תוא אלא דשרי ליה לאיניש לזרתי נפשיה
How do you know you can make a shavua to fulfill a mitzva? As it says (tehillim 119) I will swear to guard your righteous mitzvos. But surely he has already sworn at mount sinai to keep the mitzvos and this should surely be a superfluous oath? It is still a valid Neder/Shevua to motivate him to do the mitzva.
One can motivate oneself to swear to fulfill a Mitzva both with nedarim and Shevuos as is evident in Tosfos 8a:
כי היכי דאין שבועה חלה על שבועה ה״נ אין נדר חל על נדר קמ״ל דשרי ליה לאיניש לזרתי נפשיה לקיים המצוה
The Ran Nedarim 8a (vehalo) says he transgress an extra commandment of Bal yachel devoro e.g of eating pork- breaking his neder as well as the existing Torah prohibition - שהיא חלה לגבי בל יחל
Rashi Shavuos 20a also agrees - הנדרים חלים על דבר מצוה כדבר הרשות
However Tosfos Shavuos 20a and the Ramban on the torah (see gilyon Hashas nedarim 8a) in parshas Chukas says:
אם נדר לקיים מצוה כגון שאמר קונם שלא אוכל ביוה"כ אין איסור חל על איסור ולהכי איצטריך טעמא דמדרבנן
He is not transgressing the Neder as well by transgressing the mitzva, as a prohibition cannot apply on an existing prohibition from the Torah.
Accordingly his Neder is purely permitted for motivation purposes.
So the answer to the second question
is subject to debate whether transgression of the Neder would incur extra punishment of Bal Yachel on an existing prohibition Ran and Rashi say He's liable for his Neder as well, Tosfos and Ramban say he is exempt from the extra punishment of Bal Yachel.
Because some people think that a person cannot make a Shavua to fulfill a Mitzva as it is a Shavuas Shav because the Mitzva exists already, This is to prove they are mistaken:
The Gemora in Shavuos 27b that says one who makes a Shavua to transgress a Mitzva is Chayav Malkos because of Shavuas shav (he swore falsly). However if he makes a Shavua to fulfil a Mitzva he is not Chayav Malkos because of Shavuas shav whether he fulfils his oath or not as is evident above that he is motivating himself, and not making a Shavuas shav.Rambam Shavuos 5,11:
אבל אם נשבע שלא יאכל נבלה וטרפה וכיוצא בהן בין אכל בין לא אכל אין כאן חיוב שבועה כלל לא שבועת ביטוי ולא שבועת שוא:
And the Maase Rokeach explains
קי"ל בחגיגה ובמס' נדרים מנין שנשבעים לקיים את המצוות שנאמר נשבעתי ואקיימה וכו' ומעשה דבעז יוכיח שנשבע ליצר חי ה' שלא יעבור עבירה
Ramban Parshas Chukas also explains the same way:
אין חלות שאס נשבע לקיים את המצות ולא קייס אינו מתחייב
