Learn Russian: Кашель и насморк — это не грипп.

In Russian, the present tense of the verb 'to be' (is, am, are) is usually skipped. When you define something by linking a noun subject to a noun predicate (like saying 'X is Y'), you use a dash and the word это to connect them.

Since 'cough and runny nose' are two things, why don't we use a plural form of это?

When это is used as a linking word meaning 'is' or 'are', it acts as a fixed particle. It stays exactly as это, no matter if the subject is masculine, feminine, or plural.

Does the double consonant at the end of грипп change how it is pronounced?

No, the double пп is pronounced exactly like a single п. The extra letter is just a spelling rule because the word was borrowed from the European word 'grippe'.

The word кашель ends in a soft sign (ь). How do I know its gender?

Кашель is a masculine noun. Because nouns ending in a soft sign can be either masculine or feminine, you often have to memorize them. However, a helpful hint for this word is that when it changes form, the е drops out (for example, 'without a cough' is без кашля). This dropping vowel is very common in masculine nouns!