Clarify - Need to add Multiple foreign key constraints
Instead of one column per position in TeamList, you can use lookup tables for Positions and Players. The TeamList table then defines which player is in which team and in what position for each round. You then don't need to enforce integrity on all players being from the same team because there is one row per player that defines the player:team relationship in TeamList (playerID and teamID). 

I've included some DDL below as an example - you may need to add more data columns,etc. but it should be a good starting point. 

Positions table:

 CREATE TABLE [dbo].[Positions]
    (
        [PositionID] [int] NOT NULL,
        [position_name] [varchar](20) NULL,
    PRIMARY KEY CLUSTERED 
    (
        [PositionID] ASC
    )
WITH (PAD_INDEX  = OFF, STATISTICS_NORECOMPUTE  = OFF, IGNORE_DUP_KEY = OFF, ALLOW_ROW_LOCKS  = ON, ALLOW_PAGE_LOCKS  = ON) ON [PRIMARY]
    ) ON [PRIMARY]

    GO


The players table can be created as (not including team associations, but you could add this here):

CREATE TABLE [dbo].[Players](
    [playerID] [int] IDENTITY(1,1) NOT NULL,
    [player_name] [varchar](500) NULL,
PRIMARY KEY CLUSTERED 
(
    [playerID] ASC
)WITH (PAD_INDEX  = OFF, STATISTICS_NORECOMPUTE  = OFF, IGNORE_DUP_KEY = OFF, ALLOW_ROW_LOCKS  = ON, ALLOW_PAGE_LOCKS  = ON) ON [PRIMARY]
) ON [PRIMARY]

GO


You can then create your TeamList table like so 

 CREATE TABLE [dbo].[TeamList]
    (
    [TeamListID] [int] IDENTITY(1,1) NOT NULL,
    [Round(Fk)] [int] NULL,
    [Team(Fk)] [int] NULL,
    [Coach(Fk)] [int] NULL,
    [PositionID] int null,
    [PlayerID] int not null,
    CONSTRAINT [PK_TeamList] PRIMARY KEY CLUSTERED 
    (
    [TeamListID] ASC
    )WITH (PAD_INDEX  = OFF, STATISTICS_NORECOMPUTE  = OFF, IGNORE_DUP_KEY = OFF, ALLOW_ROW_LOCKS  = ON, ALLOW_PAGE_LOCKS  = ON) ON [PRIMARY]
    ) ON [PRIMARY]

    GO

    ALTER TABLE [dbo].[TeamList]  WITH CHECK ADD  CONSTRAINT [FK_TeamList_Positions] FOREIGN KEY([Position])
    REFERENCES [dbo].[Positions] ([PositionID])
    GO

    ALTER TABLE [dbo].[TeamList] CHECK CONSTRAINT [FK_TeamList_Positions]
    GO

ALTER TABLE [dbo].[TeamList]  WITH CHECK ADD  CONSTRAINT [FK_TeamList_Players] FOREIGN KEY([PlayerID])
REFERENCES [dbo].[Players] ([playerID])
GO

ALTER TABLE [dbo].[TeamList] CHECK CONSTRAINT [FK_TeamList_Players]
GO

-- Original FK definitions on round, team, coach, etc. go here