Ok. Here's the problem. In employment situations, without a contract of employment the employer can legally terminate your employment at any time, with or without cause.
So, just for purposes of termination alone, the employer could do so here. They wouldn't even need the basis for termination unless there was some suggestion that they were discriminating against you based on one of those factors I mentioned. Then they would want to have a basis for termination, to bolster that they were not making the decision based on an illegal factor (the things you mentioned about raising issues with company policy does not help here, because there is no law supporting your ability to do that. Essentially, you do so at your own risk, unless you are complaining about issues covered by Wage
and Hour law or OSHA safety
So, the employer can terminate you on the facts you've outlined here. The real issue is whether or not they can block your unemployment. Legally, they can certainly try. They have you caught in a dishonest statement. This sort of "bait and switch" questioning isn't illegal. Criminal prosecutors use it all the time and an employer has even more latitude in the manner of their questioning of employees, because they are not hindered by the criminal rules of procedure or evidence.
Unfortunately, they can certainly argue that you made an untrue statement during the course of an investigation and that that amounts to misconduct.
Now, you can certainly argue confusion on your part, in that you were more focused on the issue of the tablet and that you immediately admitted that you hadn't called the customers when the conversation actually turned to that issue. The employer would have to prove their case to block your unemployment (though they don't have to in order to terminate you), so you have a reasonable fighting chance to get that unemployment if they choose to terminate you.
In terms of what to sign and not to sign, that is difficult to say because employers don't generally have set documents that they give to employees to get them to sign. It varies greatly. You have to read each document closely to ensure that you aren't being asked to sign anything admitting misconduct or saying that you resign rather than forcing them to terminate. You also show not sign any document that waives your right to sue, unless they are giving you something in consideration for that (like severance). Finally, you don't have to sign anything in order to obtain your last paycheck, if it comes to that. They legally have to give that to you and you don't have to sign anything to get it.