Q:

Does God continually (re)create the physical world?

1) Does God continually renew the physical world in the sense that if he didn't give direct input at any given moment it would disappear?
2) If the answer to #1 is that God does continually (re)create the physical world, does he do so for the parts of the world that are not being observed by a human?

A:

Does God Continually ReCreate the World?
Yes, most explicitly according to the Shacharit prayers (ברכות ק"ש - המחדש בטובו בכל יום תמיד מעשה בראשית), R' Dessler (מכתב מאליהו חלק א), and the Zohar (חלק ב' קמט), but also according to Ramban (שמות יג:טז, דרשת תורת ה' תמימה), Bavli Chagigah 12b, and Tanya (Shaar Hayichud Vehaemunah).
No, based on extrapolations of the following opinions, all of which assume that the world runs according to "nature," ie. that there is a automatic system that God created during Creation that runs the world: Ramban (דברים יא:יג,  איוב לו:ז, בראשית יח:יט), Rambam (Moreh Nevuchim 3:18), Sefer haChinuch (546), Rashba (Responsa 1:413), and R' Saadia Gaon (Emunot v'Deot ch. 10).
 For what it's worth, it should be noted that the phrase from ברכות ק"ש appears in the siddurim of Saadia Gaon and the Rambam, and in חידושי הרמב"ן.  I don't know how they interpreted that phrase. Note: I did not check the critical editions of those siddurim, and don't know if the version I looked at is accurate. 

Re # 2:
Yes. According to that opinion, God renews the Acts of Creation (as detailed in Genesis 1), which includes everything in the universe.
 If a tree falls in the forest, and no one hears it, does it make a sound? 

A:

Nefesh Hachaim Shaar 3 Perek 1, explaining what it means when we refer to Hashem as "Makom" (literally, the Place):

ולפי פשוטו ר"ל כמו שהמקום הוא סובל ומחזיק איזה דבר וחפץ המונח עליו. כן בדמיון זה הבורא אדון כל ית"ש הוא המקום האמיתי הסובל ומקיים העולמות והבריות כולם. שאם ח"ו יסלק כחו מהם אף רגע אחת. אפס מקום קיום וחיות כל העולמות
(Translation mine) The simple idea that Chazal were conveying is that just like the place bears the object placed upon it, so too the Creator is the "Place" which bears and maintains all the worlds and creations, that if He would, Heaven forbid, remove His energy from it for even a moment, the maintenance and life-force of all creation would cease.

Similarly, the Maharal in Nesiv HaAvoda (Ch. 4) understands the word מקום to be etymologically related to מקיים, upholds.

ויורה על זה שם מקום שהוא מקיים את אשר הוא מקום לו

Additionally, the Rambam in the introduction to Chelek, 1st principle of the 13 principles, implies this:

החסוד הראשון מציאות הבורא ישתבח והוא שיש שם מצוי בשלמות אופני המציאות והוא עלת מציאות כל הנמצאים ובו קיום מציאותם וממנו נמשך להם הקיום.  ואלו נתאר סלוק מציאותו כי אז בטלה מציאות כל נמצא ולא ישארו קיימים במציאות
The first principle is the existence of the Creator, which is that there exists a perfect entity which is the cause of all that exists, and through Him their existence is maintained, and from Him their existence is drawn.  And if we would imagine the removal of His existence then all other existence would cease and nothing would remain in existence.

Aside from the indication from the words themselves, it also follows logically - if the world did not necessitate G-d's constant direct input, His hypothetical disappearance would not yield the world's cessation.

