Learn Hardcore Russian: Соль и сахар лежат на столе рядом с кружкой. - Salt and sugar are lying on the table next to the mug.

лежать = to lie (resting horizontally / just “lying there” as an object)
стоять = to stand (upright; e.g., a bottle, a glass)
сидеть = to sit (often for people/animals, sometimes things like “a hat sits on someone’s head”)
быть in the present tense is usually omitted (is/are is not normally said), so you typically choose a position verb instead: Соль и сахар лежат… = “Salt and sugar are lying (are placed)…”

Here salt and sugar (as items like a salt shaker and sugar bowl, or packets) are conceptualized as “lying” on the table.

Why is it лежат (plural), not лежит?

Because the subject is compound: соль и сахар = “salt and sugar” → two things → plural verb:

(они) лежат = “they lie/are lying”

If there were only one item:

Соль лежит на столе. (singular лежит)

But соль and сахар are singular nouns—why do they still take a plural verb together?

Each noun is singular, but the whole subject “A and B” is grammatically plural in Russian, just like in English:

Salt and sugar are… (not is) So Russian uses plural agreement: соль и сахар лежат.

Why is it на столе and not на стол?

Because на changes case based on meaning:

на + Accusative (на стол) = movement/direction “onto the table”
на + Prepositional (на столе) = location “on the table”

Here it’s a static location, so на столе.

Why is стол changed to столе?

That’s the prepositional case used after на when it means location:

стол (dictionary form, nominative)
на столе (prepositional) = “on the table”

Many masculine nouns take -е in the prepositional: в доме, на столе, в городе, etc.

What case is рядом connected with, and why is it с кружкой?

рядом (с) typically requires с + instrumental case:

рядом с кем? с чем? = “next to whom/what?” So кружка becomes instrumental singular:
кружка → кружкой And you include с: рядом с кружкой = “next to the mug.”

Could I say рядом кружкой without с?

No—standard Russian needs с here: рядом с кружкой.
(You can drop с only in some colloquial or fixed expressions with other words, but with рядом you normally keep с.)

Why is it кружкой and not кружка / кружку / кружке?

Because с (in this meaning “with/next to”) requires the instrumental case:

кружка (nominative)
кружку (accusative)
кружке (prepositional/dative)
кружкой (instrumental) ← needed after с in рядом с…

Is кружка the same as чашка?

Not exactly:

кружка = a mug (usually cylindrical, often larger, often for tea/coffee)
чашка = a cup (often smaller, sometimes with a saucer)

Either could appear in similar sentences, but the meaning changes slightly.

What does the word order mean—can it be changed?

Russian word order is flexible because case endings carry a lot of grammatical information.
Соль и сахар лежат на столе рядом с кружкой is neutral.

You can move parts for emphasis:

На столе рядом с кружкой лежат соль и сахар. (focus on location; “On the table next to the mug, there are salt and sugar.”)
Соль и сахар рядом с кружкой лежат на столе. (less neutral; can sound like you’re contrasting locations)

How would I ask “Where are the salt and sugar?” using this sentence?

A natural question is:

Где соль и сахар? = “Where are the salt and sugar?” Answer with the location:
(Они) на столе рядом с кружкой. Or keep the full verb:
Соль и сахар лежат на столе рядом с кружкой.

Does лежат mean “are lying right now” (continuous) or just “are located”?

In Russian present tense, лежат can cover both:

ongoing state (“are lying there now”)
simple location (“are (located) on the table”)

Context usually decides; there’s no separate “continuous” form like English.

If I wanted to say someone put them there, would the grammar change?

Yes, you’d likely switch to a verb of placement (perfective for a completed action), and use на стол (movement onto):

(Кто-то) положил соль и сахар на стол рядом с кружкой. = “Someone put the salt and sugar on the table next to the mug.”

For an ongoing/unfinished action:

кладёт (is putting) instead of положил.