The Excelsior Acquisition - S3-E16
Plot hole: In Series 1, Episode 7, "The Dumpling Paradox, " Sheldon makes an oblique reference to his financial status: "Frankly, if I could afford the rent, I'd ask you [Leonard] to leave, " meaning he cannot pay the rent on his two bedroom apartment by himself - not that he doesn't want to, he can't. However, in "The Execlsior Acquisition, " we find that he does not even cash his pay cheques. He doesn't even deposit them into a bank account - he leaves them in a drawer in his desk. In Series 2 Episode 14 "The Financial Permeability", he lends Penny a large amount of money from a huge bankroll he just happens to have lying about - again, without cashing his pay cheques! He is obviously independently wealthy. Either he doesn't have enough money to afford the rent or he has enough to work without being paid while practically giving large amounts of cash away. Can't be both.
The Zarnecki Incursion - S4-E19
Plot hole: If Howard can hack into Blizzard's database so deeply that he can get the name and address of game players, he could easily restore all of Sheldon's stolen 'items'. Hacking personal details - a player's real identity and their home address! - is hardcore, detailed work that only an expert could do, compared to which adding a few extra items to a player in the game would be child's play. Sheldon would not be concerned about a possible comeback from the administrators at Blizzard. First, Sheldon doesn't hack into the database, Howard does, and someone as pragmatic and self-centred as Sheldon would simply disclaim any responsibility. Second, hacking into another player's account to access highly sensitive personal information like their home address would risk incurring the same penalty from Blizzard anyway. Either Sheldon doesn't know that or he doesn't care. Third, his account has already been hacked when the virtual items were stolen. He could simply claim that Howard's hacking into his account was just more of the same - nothing to do with him. Fourth, Blizzard did not detect the hacker stealing Sheldon's virtual items, nor did they prevent it happening, and they did not detect Howard hacking into his account to obtain his name and address. Sheldon can rightly assume he has nothing to fear.
The Zarnecki Incursion - S4-E19
Plot hole: While it would certainly destroy the entire plot of the episode, Sheldon could have simply contacted Blizzard's customer support to get all his stuff back.
Suggested correction: A character not doing something that they theoretically could have isn't a plot hole. It's simply something that didn't occur to them. Regardless, the plot isn't broken because he doesn't do it; in fact, the opposite: it's required that he doesn't for the plot to work. If anything, it's a plot hook.
Suggested correction: Hacking the database to return items to Sheldon's account would also be traceable to Sheldon's account as cheating. That would get him permabanned if discovered, not a risk Sheldon wants to take.
LorgSkyegon
First, Sheldon doesn't hack into the database, Howard does, and someone as pragmatic and self-centred as Sheldon would simply disclaim any responsibility. Second, hacking into another player's account to access highly sensitive personal information like their home address would risk incurring the same penalty from Blizzard anyway. Either Sheldon doesn't know that or he doesn't care. Besides, if Blizzard were aware that Howard had hacked into Sheldon's account, why weren't they aware that the person who stole the virtual items did?
Reading information from a server is easier than writing new data into it. You can steal a mass bulk of files and extract the information at a later time, whereas writing information in Sheldon's account would take much more precision.