To answer the first question, the HaKsav VeHakaballah on Shemos 13:14 provides the context behind the 'strong hand that He took us out' reference. It writes there:
בחזק יד הוציאנו. כלומר כמו שעשה לך נפלאות למעלה מן הטבע ככה מוטל עליך, לעבדו בתורתו ובחוקותיו
With a strong hand He took us out - This means just like He made for you wonders that exceeded the realms of nature, so too he took you to serve Him with His Torah and His statutes....
So as a starting point the reference 'a strong hand' is meant to be understood that Hashem brought a miraculous salvation to save the Jewish people and we are expected to exercise our avodas Hashem with an equally dedicated fashion by performing his Mitzvos and statutes etc.
As far as the third question, the Chizkuni gives an answer on Shemos 13:16 that helps demonstrate the connection with Tefillin:
והיה לאות על ידכה תפלין של יד זכרון לחוזק יד, ותפילין של ראש זכרון לאותות ולמופתים שעשה הקב״ה לעינינו. כדכתיב בפ׳ ואתחנן ויתן אותות ומופתים בפרעה ובמצרים לעינינו.
והיה לאות על ידכה, “it shall be as a symbol on your hand” - the phylactery to be worn on the head is to serve as a symbol for the “strong hand,” with which G-d compelled the Egyptians to release the Israelites, as mentioned in the previous verse; the phylactery to be worn on the forehead however, is to be a reminder of the miracles G-d had performed as stated in Deuteronomy 6:22 - ויתן ה' אותות ומופתים גדולים ורעים בפרעה ובמצרים לעינינו, “The Lord performed signs and miracles great and awesome before our eyes against Pharaoh and the Egyptians.”
In a similar fashion refer to the Tur HaAroch on Shemos 13:9 which notes:
והיה לך לאות על ידך ולזכרון בין עיניך, “and it shall be for you a sign on your arm and a reminder between your eyes.” This means that the passages of the Torah written here shall be worn regularly on your arm and be placed between your eyes, so that you will remember all this. The reason such a parchment is also to be placed on your arm is to symbolize the חוזק יד, “the strong hand,” G’d employed in order to orchestrate your redemption from Egypt. It is a reminder of all the miracles.