"How did Judas die? Matthew 27:5 says he hung himself and Acts 1:18 says he fell and burst open."

This isn’t a contradiction since it’s possible for both accounts to be true. The additional information given in Acts is complimentary to the Matthew passage. Judas hung himself from a tree (Matthew 27:5), then could have fell and burst open on rocks (Acts 1:18) when either the branch broke or they cut him down. The skeptic must show that it’s logically impossible for both passages to be true in order for this to be a genuine contradiction. 

"Who bought the field of blood? Matthew 27:5-7 says the priests bought the field and Acts 1:16-19 says Judas bought it."

They both bought it, but in different senses. The priests bought it by physically giving money to the field owner, but technically Judas paid for it because because they used his money ("the reward for his wickedness" Acts 1:18). A similar scenario is if your grandfather died, left you money, and you bought a car with it. It would be accurate to say he bought your car by providing the money, while at the same time saying you bought the car by physically paying the dealer. The law of non-contradiction says that contradictory statements cannot both be true in the same sense at the same time. Neither in the example of Judas buying the field nor a deceased relative buying a car is this rule broken. 

"What did Judas do with the silver? Matthew 27:5 says he threw it on the ground and Acts 1:18 says he bought a field with it."

Judas threw the money on the ground (Acts 1:18) but indirectly purchased a field when the priests payed for it with his money (Matthew 27:5). This would only be a contradiction if Acts claimed that he physically gave money to the field owner in person, which it doesn’t. The law of non-contradiction says that contradictory statements cannot both be true in the same sense at the same time. Since the word "bought" can be used in different senses (especially in this circumstance), this isn’t a contradiction. A similar scenario would be if your grandfather died, left you money, and you bought a car with it. It would be accurate to say he bought your car by providing the money, while at the same time saying you bought the car by physically paying the dealer. It’s the same word used in two different senses. 

"When did Satan enter Judas? Luke 22:3,7 says it was before the last supper and John 13:27 says it was after the last supper."

Satan entered Judas on two different occasions: Once before supper (Luke 22:3,7) and once again during supper (John 13:27). On the first occasion it was to influence Judas to betray Jesus (John 13:2) and plot his arrest with the priests (Luke 22:3-4). On the second occasion it was to rally the mob to arrest Jesus (John 18:3).  

This alleged contradiction relies on the idea that Satan could only enter Judas once. However, Jesus taught that evil spirits can enter people multiple times (Luke 11:24-26). Since satan is the most powerful demonic being, it’s safe to assume he could as well. The real question is, "Why would Satan possess Judas twice when he could’ve just done it once and stayed?" Scripture doesn’t say, but we can make an educated guess. Satan indwelled Judas whenever he wanted to influence him into doing something. Since Satan isn’t omnipresent like God and can only be in one place at a time, he would have to leave if he had business elsewhere. And since his plan was to destroy Jesus, it may have required him to indwell others in order to choreograph the arrest. In order for skeptics to prove that these verses contradict each other, they must first prove that Satan could not have entered Judas on two separate occasions.