I think 'the team that is playing is correct, as 'the team' is considered as a single unit and singular, and the word 'that' is a relative pronoun which is used when referring specifically to closely related noun.
The team is playing. Joe and Pete are playing as part of the team.
In the UK the common usage in newspapers is "the team are" and "the family are". In the USA the common usage in newspapers is "the team is" and "the family is".
"You and your team ARE" is correct.
is the phrase Helen's team correct
Both MAY be correct, according to context. For example, as the object of a verb or a preposition, only "me and my team" is correct: " You saw me and my team win the game," or "The community provided support for me and my team." But as the subject of a verb, only "my team and I" is correct: "My team and I won the game."
his team.
Team Member of Winning Team
The team of professionals are experienced.
A team of girls is ("of girls" is a modifier, a team is the subject and that is singular)
The questions and issues the team are experiencing.
yes that is the correct spelling
Team management is his strength
yes
You were - This is correct. You were the best in the team.