1) Is Judaism only intellectual or is it emotional?
2) How does one feel an emotional attachment with G-d (i.e. the Shechinah) like the Jews felt at Har Sinai?
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Judaism is emotional as well as intellectual. Even the highly rationalist Rambam wrote beautiful passages about we should love and yearn for God like one does regarding a desired romantic partner. Early mussar seforim, such as Chovos HaLevavos, are focused to a large extent on developing intense love for God. The Ramak's Tomer Devorah is full of references to the importance of being full of love for God and all His creatures.
As for how to feel an emotional attachment to God, a variety of ways have been explained, from the earliest mussar seforim to more recent Chassidic philosophy. The encyclopedic volume Jewish Spiritual Practices by Rabbi Yitzhok Buxbaum is a wonderful resource for Chassidic teachings on this general subject (and also includes some classical and Litvish sources as well, such as many quotes from the Litvish mystic Rabbi Alexander Zyskind.)
I am not sure whether any of these sources talk of reaching the level of how Jews felt at Har Sinai, but experiencing the Glory of God is sometimes mentioned in mystical sources. The Tanya describes at length the process of awakening an all-consuming love for God, including through contemplating His greatness in particular ways. The Bilvavi (Rabbi Itamar Schwartz) discusses how to cultivate love of God, based in part on the works of Rabbi Alexander Zyskind.
Judaism demands both intellectual (atah haresa lada'as) and emotional (tachas asher lo avadata es hashem elokecha bisimcha uvituv levav) engagement.
Reaching the level of feeling of Har Sinai may not be entirely reasonable as a daily practice to engage in for the average individual. However, anyone may feel an emotional attachment to Hashem by reading and concentrating on the words of tefillah and bringing them into his heart, thinking about the wonders of the creation and all that Hashem does and continues to do for us.
"Talking" to Hashem on a regular basis outside of tefillah is a very easy way to feel attachment to Hashem. I find that stopping for a few seconds to close my eyes and my mind to the world and concentrating while saying/thinking repeatedly "I love you" is probably the most powerful way to jump start that feeling of love. Just like it's also a powerful thing to do in a relationship.
Probably the hardest part of prayer is drawing the emotional feeling from the heart, which is why it is called an avodah shebalev. As an actor, I found that approaching tefillah as a script was very conducive to developing my ability to engage in it with emotion. Bow here, say this, move there, sit down. Anyone can read the words of a script, but as an actor (Jew), your job is to take those actions and lines (prayers) and invest them with as much reality and meaning you can. When you manage to accomplish that, the emotion naturally flows.
As always, there will be good and bad days, performances and prayers where your head or heart "wasn't in it," or "the energy wasn't there," or where you flub a line in what was a perfect "show." The key is to keep pushing forward, keep practicing, and keep in mind how valuable and important what you are doing is.