It sounds like an unfortunate situation you have on your hands. Yet, without more information it is difficult to determine what your course of action should be regarding the notice to abate. However, here are a few bits of information that might be helpful.
1) Most localities have a procedure for appealing the finding of a public nuisance. Indeed, the failure to provide an opportunity to challenge the finding of a public nuisance could raise constitutional concerns. The procedure for appealing the finding of a public nuisance is usually laid out in the city code and often has very quick deadlines for filing an appeal. You should find out what those deadlines are or consult with an attorney about your options as soon as possible.
2) The sheriff had the right to enter your property for the purpose of knocking on the door and posting the abatement notice regardless of any signs you may have posted. Likewise, the sheriff has the right to cross through “open fields” on your property in spite of your no trespassing signs. However, the Fourth Amendment to the Constitution generally requires that he obtain a warrant before he can inspect the area immediately adjacent to your house. If he failed to do so, then your rights have been violated.
3) The city has the authority to require you to remedy conditions on your property that endanger the public health and safety. If those conditions exist on your property you should probably request time to remedy the situation rather than fighting the abatement notice. However, city officials can be wrong about whether or not your property is a public health and safety concern. If so, you are completely within your rights to fight the public nuisance finding and should contact an attorney to determine your options.
It is not exactly clear what they cited you for. There is a city ordinance that you have to abide by which regulates certain exterior & upkeep features. For example, you can be cited for having high weeds in your front yard. Seems like they entered your home so it may be that there's something unsafe there ... You need to provide additional information, so that an attorney can comment in more detail.
Check with your city to determine what process is available for disputing abatement notices. If there is something on your property that could be a public nuisance, you may be better off asking for time to abate it, rather than fighting it and just delaying the inevitable.
The city inspectors have certain rights to perform inspections. Your no trespassing sign is not sufficient to actually keep someone off the property, but might give you grounds to bring a lawsuit for the violation.
Generally, if the inspector was performing his legal duty, any lawsuit for trespassing would probably not be successful. Government employees have immunity from most lawsuits.
Contact a local real estate attorney. Realistically speaking, the least expensive avenue is going to be cleaning up the nuisance. If you fight and lose, you may end up paying the city to clean it up.