Learn Hardcore Czech: Pro moji babičku je tato nemoc nejhorší. - For my grandmother this illness is the worst.

Mojí babičce is dative (to/for my grandmother), but that form is used without the preposition pro, in different sentence types, for example:

Moji babičce je 80. = My grandmother is 80 (years old).
Pomáhám mojí babičce. = I help my grandmother.

In your sentence, because of pro, accusative babičku is required.

Can I also say Tato nemoc je pro moji babičku nejhorší? Is it the same?

Yes, Tato nemoc je pro moji babičku nejhorší is completely correct and very natural.

The difference is mainly word order and emphasis:

Pro moji babičku je tato nemoc nejhorší.
– Emphasis starts from her perspective: For my grandmother, this disease is the worst (maybe compared to other people).
Tato nemoc je pro moji babičku nejhorší.
– Emphasis starts from this particular disease: This disease is the worst for my grandmother (maybe compared to other diseases).

Both are grammatically fine; Czech word order is fairly flexible, and speakers choose the order depending on what they want to highlight.

Why is it tato nemoc and not just nemoc or ta nemoc?

All three are possible, but they have slightly different nuances and styles:

tato nemoc – more formal / neutral, similar to this illness in standard written English.
ta nemoc – a bit more neutral/colloquial, like that illness / that disease (often something already mentioned in context).
tahle nemoc – clearly colloquial / spoken, like this illness here (more emotional or casual).

If you said only nemoc without any determiner, it would usually refer to illness in general:

Nemoc je pro moji babičku nejhorší.
This sounds like Illness is the worst thing for my grandmother (in general), not about one specific disease.

Your sentence with tato nemoc clearly points to one specific illness you have in mind.

What does nejhorší literally mean, and how is it formed?

Nejhorší is the superlative form of the adjective špatný (bad).

The comparison pattern is irregular:

špatný = bad
horší = worse
nejhorší = worst

So nejhorší literally means the most bad / the worst.

The form nej- + horší is a typical way of forming superlatives: you take the comparative (horší) and add nej- to the beginning:

dobrý → lepší → nejlepší (good → better → best)
mladý → mladší → nejmladší (young → younger → youngest)

Does nejhorší change its form for gender and number?

Yes, in Czech adjectives normally agree with the noun in gender, number, and case. However, the form nejhorší looks the same in several genders and numbers, which can be confusing.

For nemoc (feminine singular), nominative is:

ta nemoc je nejhorší

Other examples:

ten den je nejhorší (masculine inanimate singular – same form nejhorší)
to auto je nejhorší (neuter singular – same form nejhorší)
ti lidé jsou nejhorší (masculine animate plural – also nejhorší)

So yes, it agrees with nemoc, but in this particular form, you do not see the change on the word itself – it just happens that nejhorší is the correct ending here for feminine singular nominative as well.

Why do we need the verb je? In English we sometimes drop is in headlines; can we drop je here?

In normal Czech sentences, you must use the verb je (the present form of být – to be):

Pro moji babičku je tato nemoc nejhorší.

Leaving out je would sound incomplete or like a title/headline, for example in a newspaper or on a poster:

Pro moji babičku tato nemoc nejhorší – acceptable only as a very stylized headline, not as normal speech or writing.

So for regular communication, always include je.

Is pro moji babičku about giving something to her, like for my grandmother to receive?

No, here pro moji babičku expresses point of view / impact, not a future recipient.

Some uses of pro:

Recipient / destination
- Dárek pro moji babičku. = A present for my grandmother.
Point of view / evaluation (your sentence)
- Pro moji babičku je tato nemoc nejhorší.
  = From my grandmother’s point of view, she suffers the most from this illness / this illness affects her worst.
Purpose / intended use
- Lék pro pacienty. = Medicine for patients.

In your sentence it clearly means from her perspective / in her case, not intended for her as a gift.

Is pro mou babičku more correct than pro moji babičku?

With a feminine noun like babičku, both are possible in modern Czech, but there is a stylistic difference:

pro mou babičku – traditionally considered more standard / literary / formal
pro moji babičku – very common in everyday speech, often felt as more colloquial

In careful written Czech (e.g. school essays, official texts), teachers might prefer pro mou babičku.
In spoken Czech, pro moji babičku is extremely frequent and sounds natural.

So grammatically you are fine with both; be aware of the slight formality difference.

Why does babičku end with -u here?

Babička is a regular feminine noun of the -a type:

Nominative: babička (who? what?)
Accusative: babičku (whom? what?)

The preposition pro requires the accusative, so:

pro babičku
pro moji babičku

This -u ending is typical for feminine -a nouns in the accusative singular:

maminka → maminku (mom)
sestra → sestru (sister)
teta → tetu (aunt)

How would the sentence change if I wanted to say my grandmother handles this illness the worst (adverb), not this illness is the worst for her (adjective)?

Your original sentence uses nejhorší as an adjective describing the illness:

Pro moji babičku je tato nemoc nejhorší.
= Among all illnesses, this illness is the worst one for her.

If you want to say that she copes the worst (her reaction is the worst), you use the adverb nejhůř:

Moje babička snáší tuto nemoc nejhůř.
= My grandmother tolerates / handles this illness the worst.

So:

nejhorší – adjective, describing a thing (the worst illness)
nejhůř – adverb, describing how someone does something (handles it the worst)