It's right in front of you!

The tree-house is 'up' off the ground, yet still 'under' the main body of the tree, so these terms consecutively make good sense. Even if the tree (and house) is downhill as seen from the main building itself, 'down under' (Answer 2) does not sound right.
The construction of the tree-house would indeed have involved moving and installing parts upwards and into the tree, but English would not bother with the 'into' preposition (Answer 3; confirming a sense of motion, as in where German would use the Accusative Case in preference to the Dative); while 'way up' (Answer 4) sounds rather too high for safety, and would need to be expressed as 'way up in the tree'.
The second-blank Answers were all mostly reasonable but only Answer 1 offers a good pair.

It might even have been 'left out' (i.e. not put away, or projecting across where people may need to walk), but no such Answer was offered. The bike may indeed have been left, right in the middle ~ odd though the double pun may seem!

Any of the word-orders other than Answer 4 will do the trick fine here.

Answer 1 is too vague, Answer 3 doesn't make geometrical sense, and 'over' (Answer 4) suggests somewhere further away than the context leads us to expect (the machine is nearby, yet the switch is 'over there' ... beyond the machine perhaps, rather than just behind it?).

The package (presumably a terrorist bomb) was LEFT ( = abandoned, prior to it detonating; Answer 1), RIGHT (Answer 2) in the precise spot where its blast would inflict most damage, or thereabouts (Answer 3); nobody with the commitment to do such a thing would set about it 'vaguely' (Answer 4), for sure!

The first three Answers are acceptable (No.3 perhaps fractionally less good), but Answer 4 suggests a mode of distribution more like the spreading of a single, solid or viscous layer (e.g. a blanket, or butter and/or jam on a slice of toast), rather than that individual children or small groups would be 'dotted' or 'stippled' (good word: look it up!) through the grounds.

The point of the adverb here is to emphasise the shortness of distance, rather than the precision of the directions!

'Just' can (originally) mean 'having the quality of justice' (as in Answer 1) or, adverbially, 'merely' (Answer 3) or 'virtually' (Answer 2). We do have a phrase 'Just a moment' (Answer 4), but that does not make relevant sense in the context.

Answer 1 is really quite vague; Answer 2 clearly enough suggests your destination is not on a very high floor; Answer 3 suggests that it is 'far, but not all that far' from the top (perhaps within the range of 25 to 15% below the top); but 'quite near' (Answer 4) appears to suggest a range just that little bit closer still. After all, it does not contain any contradictory negative element.

These are all good English noise words (including the breaking glass in Answer 3), but for some reason the standard version is Answer 2.