I'd take out the "of". Any of these would be equivalent: Tom's giving a book to Jack [...] Tom's giving Jack a book [...] Side note: without the rest of the sentence "Tom's" sounds more like a contraction (Tom is) than a possessive noun (belonging to Tom).

In this case, I think "some juice" and simply "juice" would be semantically equivalent. I don't sense much difference between these: Is he looking for juice? Is he looking for some juice? If a linguist insisted, I'd explain the difference similar to how you did with one correction. Compare these two questions: 1. Is someone there? 2. Is anyone there? Like you say, in 1 the "some" anticipates an affirmative answer. You more than 50% expect the answer to be yes. The correction I'd make to your explanation is that you *expect* the answer to be yes, but you don't necessarily *want* it to be yes. In 2, you would more than 50% be expecting the answer to be no, even if you wanted it to be yes. If the textbook is presenting that sentence as an example of the some/any opposition, the unmodified equivalent (Is he looking for juice?) would be the more uncertain version. Having said all that, yeah, I think other native speakers wouldn't put this much thought into it, and they would or wouldn't use "some" interchangeably in this case.

In case there's any misunderstanding, "downtown" doesn't always mean "in the centre of the city". Downtown is just wherever there's the highest concentration of people, commercial activity, etc. in a given city, and that can be on its edge in some cases.

I would say "The big shops are downtown" or "The big shops are in the centre of the city", depending on whether the person you were talking to understood where downtown in this city was located. "The big shops are downtown" is probably the more natural of the two.

The way the author has phrased this is a bit too clever for my liking, but here's what they mean: The bacterium has to be able to experience more than one moment in time at once (that is to say it has to have some equivalent to short term memory) so it can compare the current conditions to past conditions. It's moving through a single region of space at a time but constantly comparing more than one instance in time, even if it's just a moment ago: Is the concentration of what I'm looking for higher now (in this space) or was it higher a moment ago (in that spot where I no longer am)?

Off the top of my head, you can use 1. the modal verb "will" before a verb in the infinitive without "to" 2. "am/is/are going" before a verb in the infinitive with "to" 3. the present or present continuous tense, often with a modifier to show that it will happen in the future: 1. I *will* visit my dad tomorrow. We *will* have an argument, I'm sure. 2. I *am going* to eat the entire cake once it has finished baking. I *am going* to regret it, but that *is not going* to stop me. 3. I *am taking* the kids to the lake Sunday, but first we *are having* a giant junk food breakfast. There are others (e.g. future perfect: "I will have eaten all the cake before they arrive"). Check Chris's link for more detail.

You could simply say "through": New check-ins are suspended through May 31. Another common phrasing to remove any ambiguity would be "up to and including": New check-ins are suspended up to and including May 31.