При сильном ветре я не езжу на велосипеде.

При literally means “at/under” (a condition). In sentences like При сильном ветре…, it means “under conditions of…” → “when/if there is strong wind.”
It sounds a bit more “conditions-based” (and sometimes slightly more formal) than just saying когда.

При governs the prepositional case.
So ветер → prepositional (о) ветре / при ветре = ветре.

Adjectives must agree with the noun in case, gender, and number.

• ветре = masculine singular prepositional
• therefore сильный → masculine singular prepositional = сильном
  So: при сильном ветре.

Yes, both are common, but they’re not identical:

• при сильном ветре = “under conditions of strong wind” (condition/circumstance)
• в сильный ветер (accusative) = more like “in strong wind” (situation/background), often very natural for weather

Both can translate as “when it’s windy,” but при emphasizes the condition.

Because ездить / езжу is used for habitual, repeated, or general actions (“I don’t (generally) ride”).
ехать / еду is for one specific trip happening now or planned now (“I’m not going (by bike) (right now / this time)”).
So:

• При сильном ветре я не езжу… = general rule/habit
• Сейчас сильный ветер, я не еду… = this particular occasion

езжу is the 1st person singular present of ездить.
It’s irregular in form: the consonant changes in conjugation:

• ездить → езжу, ездишь, ездит…

This is just a memorized present-tense pattern.

Not strictly. Russian often omits subject pronouns because the verb ending already shows the person.

• При сильном ветре не езжу на велосипеде. = perfectly natural
  Adding я can add a bit of emphasis/contrast: “I (personally) don’t…”

For “by bike / riding a bike,” Russian most commonly uses на + prepositional:

• ехать/ездить на велосипеде

велосипедом (instrumental without на) is possible in some contexts, but на велосипеде is the standard everyday phrasing for transportation.

In на велосипеде, the preposition на is used with the prepositional case for location/means in this pattern.
So велосипед → (на) велосипеде.

не normally goes directly before the word it negates—here the verb: не езжу.
You can rearrange other parts for emphasis, but не + verb typically stays together:

• Я на велосипеде при сильном ветре не езжу. (emphasis on “by bike”)
• При сильном ветре я не езжу на велосипеде. (neutral)

Yes, the ending changes and the stress stays on the first syllable:

• ве́тер (VE-ter)
• в ве́тре / при ве́тре (VYE-tre)

And велосипе́де has stress near the end: ve-lo-si-PE-de.