Planet Hollywood is the most south with easier access to Cosmopolitan, Bellagio and MGM Grand while the Flamingo is in between both with easiest access to O'Sheas, Caesars Palace and the Forum Shops. We prefer both the north and south because we can get to the newer places on The Strip more quickly.
The rooms are hit and miss with Planet Hollywood having the most recently updated rooms and Harrah's rooms being the oldest. Again, Flamingo is in the middle with some rooms being updated, but not all. We like new rooms, but we're OK with all your choices. None are great, none are awful. Just a heads up, Caesars housekeeping is spotty across the board.
All of the hotels you're considering are close to decent food options, but we like the choices at Planet Hollywood best. There are multiple places to get coffee and cheap sandwiches @ Earl of Sandwich as well as nicer high-end selections.
The casinos are similar with middle tier limits at the tables ($5-$15) and a wide array of slot machines and mediocre video poker pay tables. The main difference is that the Planet Hollywood casino gets super loud on weekend nights, so just be prepared for that if you stay there.
On to loose slots, Palms claims to have looser slots than The Strip properties and while most off Strip casinos are slightly looser (1-3%), there isn't much of a difference. Just know that the higher denomination you play, the better payout percentages are. So penny slots pay the least back, while $1 slots pay better.
One good thing about the Caesars properties is that they all have a decent selection, but the MGM Resorts properties have a newer selection. The off-Strip properties are hit and miss depending on size, but Palms usually has the new shiny slots.
We like staying at Planet Hollywood because we enjoy staying with a younger crowd, having cheap food options and quick access to The Cosmo and Bellagio, which are among our favorite spots on The Strip.
When it comes to slots Palms, Red Rock, and Main Street Station may be better to extend your play by a few minutes, but the overall effect isn't much. We are Strip dwellers when we stay there, but leaving will certainly extend your play.
Let us know where you stay and how it was. Have a great trip!