Before I beginstart, I would really appreciate suggesstions in terms of writing style. Before I start, I would really appreciate suggestions in terms of writing style.

start is another way of saying it, but begin is still correct :)

Before I begin, I would really appreciate suggesstions in terms of writing style. Before I begin, I would really appreciate suggestions in terms of writing style.

BetweeOn Wednesday and Thursday, Germany experienced the influence of an orcan called Ylenia. On Wednesday and Thursday, Germany experienced the influence of an orcan called Ylenia.

between is correct but since it's only 2 days "on" would be a better way to say it

BetweenFrom Wednesday andto Thursday, Germany experienced the influence of an orcan called Ylenia. From Wednesday to Thursday, Germany experienced the influence of an orcan called Ylenia.

Citizens were asked to stay indoors and schools were closed statewide in NRW, East Germany. Citizens were asked to stay indoors and schools were closed statewide in NRW, East Germany.

Citizens as it is a group of people, "indoors" makes it more specific to mean inside

Citizen were asked to stay inat home, and schools were closed statewide in NRW, East Germany. Citizen were asked to stay at home, and schools were closed statewide in NRW, East Germany.

Compared to the past, the states can now act by themselves and close schools. Compared to the past, the states can now act by themselves and close schools.

adding "by" is a more natural way of saying it

Compared to the past, the states can now act themselves andUnlike before, the states now have the authority to close schools themselves. Unlike before, the states now have the authority to close schools themselves.

Alt: "Compared to before"

However, after the situation calmed down, I traveled to school to take an exam. However, after the situation calmed down, I traveled to school to take an exam.

Alt: "went" instead of "traveled"

On my way, I saw trees who felt downfallen trees - one on the motor way which was removed by officials. On my way, I saw fallen trees - one on the motorway which was removed by officials.

"fallen" can be used as an adjective instead, motorway is normally one word

On my way, I saw trees who felt down - onethat had fallen down. One was on the motor way, which was then removed by officials. On my way, I saw trees that had fallen down. One was on the motor way, which was then removed by officials.

Unfortunately, the train iI was going withon made longer stops, resulting in a longer way home. Unfortunately, the train I was going on made longer stops, resulting in a longer way home.

"I" is capitalised when talking about oneself, we use "on" when talking about travelling by train, you don't have to add "er" to "long" but it can be added for emphasis

Unfortunately, themy train i was going with made longer stops, resulting in a long waywhich increased my commute time home going home. Unfortunately, my train made longer stops, which increased my commute time home going home.