The problems are all plural - you must decline the nouns in the plural form. It is not incorrect to put 'today' first, but it would sound more natural after 'world'. If it is first, it must be followed by a comma.
Traffic accidents [is] [are] one of [the] [these] serious problems that often happen unintentionally[,] and sometimes [this] [they] can result in personal injury, property damage or [even] death [as well - DELETE].
Again, remember these are plural traffic accidents. Also, your final clause is not incorrect - I have modified it to emphasise 'death' because it sounds more natural that way.
In Phnom Penh, a lot of people die or [injure] [are injured] because of traffic accidents and this problem [becomes] [is becoming] more and more serious [due to] [for] many reasons. The first and [most] important reason is caused by - DELETEdrunk drivers. In general, drunk drivers can not control their [feeling] [emotions] and they usually drive fast and carelessly. [With this action] [Because of this] car accidents will - DELETE (should be present tense) happen easily. Another [cause] [reason] is that people do not obey the traffic laws. Some people do not care about traffic lights. For example, they may [go through the streets] [carry on] even though the traffic light is red. In addition to traffic lights, in some situations [people] may also cause road accidents [such as] [by] looking for something in the car, using [a] mobile phone, driving fast at [the] [a] crowded location or sleeping during driving.
- For Teachers