Learn Hardcore Chinese: tā zài shǒujī shàng kàn dìtú, xiǎng zhǎodào gōngyuán de nán mén. - He is looking at the map on his phone, trying to find the south gate of the park.

In Mandarin, location phrases with 在 usually go before the main verb:
他在学校学中文。 – He studies Chinese at school.
So 在手机上 (on his phone) naturally comes before 看地图 (look at the map).
Putting it after the verb, like 他看地图在手机上, sounds unnatural or at least very marked.
The normal pattern is: Subject + 在 + Place + Verb + Object → 他在手机上看地图.

What exactly does 手机上 mean? Why is 上 used with 手机?

手机上 literally means on the phone, referring to the phone’s screen or interface.
In Chinese, 上 is often used for surfaces or platforms, including digital ones:

桌子上 – on the table
墙上 – on the wall
网上 – on the internet
By extension, 手机上 means on (the screen of) the phone, not physically on top of it in a literal way.

What’s the difference between 在手机上看地图 and 用手机看地图?

Both can be translated as look at a map on the phone, but the nuance is slightly different.

在手机上看地图 emphasizes the location / medium: the map is displayed on the phone.
用手机看地图 emphasizes the tool: he is using the phone to look at the map.
In everyday speech, both are common and natural; the choice is more about what you want to highlight: place vs tool.

Why is it 看地图 and not something like 看着地图 here?

看地图 simply states the action: to look at the map.
看着地图 adds the feeling of being in the state of looking (more continuous, descriptive).
For a neutral narrative sentence describing what he is doing, 看地图 is the default.
If you wanted to emphasize that he is busy looking and remains in that state, you might say:
他正看着地图，没注意到我。 – He was looking at the map and didn’t notice me.

What is the function of 想 in 想找到公园的南门? Is it “want to” or “think”?

Here 想 means to want to / to intend to / to try to, not to think.
想 + Verb often expresses a desire or intention:

我想睡觉。 – I want to sleep.
In this sentence, 想找到 means he is trying / hoping / wants to find the south gate.
If you wanted think, you would usually see 想 followed by a clause, like 我想他已经到了。 – I think he has already arrived.

What’s the difference between 找 and 找到 in this sentence?

找 is to look for, focusing on the process.
找到 is 找 + 到, where 到 is a result complement indicating success / arrival at a result.
So:

找公园的南门 – to look for the south gate (maybe succeed, maybe not).
找到公园的南门 – to find / to manage to find the south gate (focus on actually finding it).
Here, 想找到 shows his goal is not just to search, but to end up finding the gate.

Why is it 公园的南门 and not 南公园门 or some other order?

The basic pattern is “larger thing” + 的 + “part of that thing”.
公园 (park) is the larger place; 南门 (south gate) is a gate belonging to that park.
So 公园的南门 literally means the south gate of the park.
Chinese normally puts the owner / whole before 的, and the part / attribute after it:

学校的图书馆 – the school’s library
中国的首都 – China’s capital
So 公园的南门 follows this same pattern.

Is 的 in 公园的南门 always necessary? Can I say 公园南门?

You often hear 公园南门 in everyday speech, especially in short noun phrases (signs, directions, etc.).
Both 公园的南门 and 公园南门 are acceptable here.
Using 的 sounds a bit more explicit / neutral / slightly formal; dropping 的 makes it shorter and a bit more “headline-like”.
As a learner, you’re safe keeping 的; later you’ll get a feel for when it can be omitted.

Does this sentence mean he actually found the south gate, or just that he wants to find it?

This sentence only states his intention or goal, not the outcome.
想找到 = wants to / is trying to find (and implies success is hoped for, but not confirmed).
To say he already found it, you’d say something like:
他在手机上看地图，找到了公园的南门。 – He looked at the map on his phone and (successfully) found the south gate.
The original sentence stays at the “trying / aiming to find” stage.

How is 找到 (zhǎodào) pronounced with tones? Do both syllables stay third tone?

Underlyingly it’s zhǎo (3rd tone) + dào (4th tone).
Chinese tone sandhi mainly affects 3rd + 3rd sequences; here it is 3rd + 4th, so no change:
You pronounce it as zhǎodào, with a dipping 3rd tone then a falling 4th tone.
So: 找 stays a full 3rd tone, 到 stays a full 4th tone.

What does the comma ， do in this sentence? Is it like “and”?

The comma separates two closely related clauses:

他在手机上看地图 – He is looking at a map on his phone
想找到公园的南门 – (he) wants to find the south gate of the park
The subject 他 is understood for the second clause and not repeated.
The comma does not literally mean and, but in English we often translate it as and or so that / in order to depending on context.
Here you could read it as: He is looking at a map on his phone, trying to find the south gate of the park.

Is 想找到公园的南门 expressing purpose, like “in order to find the south gate”?

Yes, that’s a very natural way to understand it.
The second clause gives the reason / purpose for the first action.
More explicit purpose markers like 为了 could be used:
他在手机上看地图，是为了找到公园的南门。 – He is looking at the map on his phone in order to find the south gate.
In everyday speech, just using 想 + Verb after a comma is often enough to convey purpose.

Could we say 他在手机上找公园的南门 instead of 看地图，想找到公园的南门?

Yes, that would also be a natural sentence, but it changes the focus slightly.

在手机上看地图，想找到公园的南门 emphasizes the action of looking at the map and his goal.
在手机上找公园的南门 emphasizes that he is using the phone (e.g. map app) to search directly for the south gate.
The original highlights the intermediate step (look at a map) plus the intention (to find the gate).