I want to get this right.
The stages sequence runs like this. -Game is played- enter game stage(24 hours)(At this point my team home page will read "Midweek in xx(time)"). -enter midweek stage(24 hours)- next game is played.
Is this correct??
2nd question.
I have 53 players on contract. During midweek I can sign a free agent putting me at 54 players, but I must cut a player during the next game stage to get back to 53. Is this correct??
If I sign a player during game stage I must also cut a player during this same game stage. Is this correct??

Yes, that is all correct. The stage actually runs for about an hour (sometimes more, sometimes less) when the countdown clock hits zero - during this time you won't be able to access the league. The game is more or less "frozen" between stages to allow you to prepare for the next stage, so you're not actually "in" a stage, you are getting ready for the next stage.

Your 2nd question is correct, that's the main purpose of having the midweeks. One other item of note is that if you sign a player during a game stage (and cut another to make room) the new player will be added as an _inactive_ player, so you'll want to make sure that your depth chart is prepared to handle not having the player you cut and not expecting your new player to be in the depth chart. If you have fewer than 45 active players on game day the AI will rebuild your depth chart. Again, a very good reason to make these roster changes during the midweek, so that you can go into the game stage with complete control over the team you are fielding.

Sorry but I want to just clarify this.
Game is played(1 hour)- now getting ready for midweek(23 hours)- midweek stage(1 hour)- getting ready for game stage(23 hours). game stage(game is played)( 1 hour). This is more exact, right??

Now its during the "getting ready for midweek" I can offer the FA a contract. then midweek stage is played and I now hopefully have signed my FA. I have 54 players. and we are getting ready for game stage. here I cut my player. Game stage hits. my player is cut. I am back to 53 players and the game is now played. Whew. is this correct?? I Don't mean to be annoying but I want to make sure I have this right.

No worries - I would rather you understand than be frustrated because you don't!

Your scenario you describe is exactly how it would play out.