Learn Hardcore Japanese: natu ni kazoku to issyo ni nasi wo tabemasu. - In summer I eat pears together with my family.

The full version could be: 夏に私は家族と一緒になしを食べます。
(In summer, I eat pears together with my family.)
But since the sentence is about something you do, 私は is understood and normally omitted.

So there is an invisible “I” as the subject:
(私は) 夏に家族と一緒になしを食べます。

Why are there two に in the sentence (夏に and 一緒に)? Don’t they mean the same thing?

They are the same particle に, but used in different roles:

夏に
- に marks a time point/period.
- 夏に = in summer / during summer.
一緒に
- 一緒に is almost like one unit meaning together (with).
- Here に is attached to 一緒 to make the adverb together.

So:

夏に → time: in summer
一緒に → manner: together

Why do I need both と and 一緒に in 家族と一緒に? Could I just say 家族と or just 一緒に?

家族と一緒に literally means together with (my) family.

家族と = with my family
一緒に = together

Using both emphasizes the idea of doing it together with them.

You can say:

家族となしを食べます。
→ I eat pears with my family. (Perfectly natural; 一緒に is just not said.)
家族と一緒になしを食べます。
→ I eat pears together with my family. (Slightly more explicit/emphatic about “together”.)

But you cannot say:

家族一緒になしを食べます (missing the particle と, ungrammatical)
一緒になしを食べます。 without saying who you’re together with — that’s fine, but then it just means I eat pears together (with someone / with others), and who is together is left to context.

So 家族と一緒に is a very common set phrase meaning together with (my) family.

Can I change the word order, like 家族と一緒に夏になしを食べます or なしを夏に家族と一緒に食べます?

Yes, Japanese word order is fairly flexible as long as the verb stays at the end and particles stay attached to the right words.

All of these are grammatically OK and mean the same thing:

夏に家族と一緒になしを食べます。
家族と一緒に夏になしを食べます。
夏になしを家族と一緒に食べます。
なしを夏に家族と一緒に食べます。

The most natural and neutral version is probably the original:

夏に家族と一緒になしを食べます。

Changing the order can slightly affect emphasis (what feels “in focus”), but not the basic meaning.

Why is it なしを食べます and not なしが食べます? What does を do here?

を marks the direct object of the verb — the thing that the action is done to.

なし = pear(s)
食べます = eat
なしを食べます = eat pears

Using が here would make pears the subject:

なしが食べます。
→ Pears eat (something). (This would mean the pear is doing the eating, so it’s wrong in this context.)

So:

Xを食べます = eat X
を = object marker

What tense is 食べます? Does it mean I eat, I will eat, or I am eating?

Japanese -ます form without ている is a non-past form. It covers present and future.

So 食べます can mean:

I eat (generally / habitually)
I will eat / I am going to eat

The exact meaning is decided by context.

In a simple sentence like:

夏に家族と一緒になしを食べます。

it is usually interpreted as a regular or habitual action:

I (usually) eat pears with my family in summer.

If you were clearly talking about a future plan, it could also be:

This summer I will eat pears with my family.

Do I need to use a counter word for pears? Is なしを食べます enough?

なしを食べます is fine if:

you don’t care about the number, or
the number is obvious from context.

If you want to say how many pears, you add a counter:

なしを一つ食べます。 – I eat one pear.
なしを二つ食べます。 – I eat two pears.
なしを三つ食べます。 – I eat three pears.

You can also put the counter elsewhere:

夏に家族と一緒になしを三つ食べます。
– In summer, I eat three pears together with my family.

So:

No number or unimportant number → なしを食べます
Specific number → なしを三つ食べます, etc.

Does なし always mean “pear”? I’ve seen なし mean “none / without” too.

There are two common words pronounced なし:

梨（なし） – pear (the fruit)
無し（なし） – none, without

In your sentence:

夏に家族と一緒になしを食べます。

it must be 梨 (pear), because:

It takes を and is being eaten.
The meaning “eat none / eat without” doesn’t work here.

In writing, context and kanji separate them:

梨を食べます。 – I eat pears.
お金は無しです。 – There is no money.

What does 家族 exactly mean here? Is it “my family” or just “family” in general?

家族 literally means family, but:

When you say 家族と～ about your own actions, it is understood as my family.
Japanese often omits “my/your” if it’s clear whose something is.

So:

家族と一緒になしを食べます。
→ I eat pears together with *my family.*

If you needed to be more explicit, you could say:

私の家族と一緒になしを食べます。
But this is usually unnecessary unless you want to contrast it with someone else’s family.

Could I say 夏には家族と一緒になしを食べます instead of 夏に? What’s the difference between に and には here?

Yes, you can say:

夏には家族と一緒になしを食べます。

Here は is the topic / contrast marker added to 夏に.

Nuance:

夏に家族と一緒になしを食べます。
→ A simple statement: I eat pears with my family in summer.
夏には家族と一緒になしを食べます。
→ Slight emphasis on summer, often with a feeling like:
As for summer, (at least) I eat pears with my family.
It can imply a contrast such as:
- In summer, I eat pears;
- in winter, I eat something else, etc.

So には = に + は, making 夏 the topic / point of contrast.

When should I use 食べます vs 食べる? And do I ever say 食べますです?

食べます is polite form.
食べる is plain / casual form.

Use:

食べます with:
- people you don’t know well
- at work, in formal situations
- in most textbooks and beginner materials
食べる with:
- close friends
- family (often)
- when speaking casually, in diaries, etc.

Examples:

夏に家族と一緒になしを食べます。 – polite
夏に家族と一緒になしを食べる。 – casual

You never say 食べますです.
In this kind of sentence, ます already carries the politeness; you don’t add です to the verb.