Ok, I think I'm going to give up my company name (see here: Trademark infringement - does it really sounds that bad? )
Given my situation I need to make sure:
I found a pretty unique name. It's actually a latin word, it has good abbreviation, it sounds like the previous name, it will be somewhat memorable. It is 100% unique in the trucking/transportation industry, which is what I need. Even parts of it have not been used.
Can you imagine a problem trademarking such a name later on? My lawyer says it should not be a problem because businesses are different and there's no correlation, but who knows?
Do you see a problem with using the .NET domain name?
I primarily work with US companies and in the US it's instinct to add .com after anything you type in as a web address. You could have just heard the person say www.whatever.net and you'll type in www.whatever.com - it's that ingrained in our behavior. SO - I only buy .com's.
So... I would suggest adding something to your domain. So if your company is Lorem and you can get Lorem.net but you are in the trucking business buy LoremTrucking.com or LoremApp.com
The App after the name is getting pretty common for software and mobile companies.
Like 37signals wanted something simple for their product name 'Basecamp' so they got BasecampHQ - not a big deal... they still call their app something one word but their domain is a little bit longer.
Your second issue, regarding the trademark, should be addressed sooner rather than later. Apply for the trademark once you have decided you are going with the new name.
If you are worried about infringing on the police department (which seems unlikely as you are probably in very different marketplaces) why not call them up and ask if they are likely to object? It's not legally binding but it might save you some money pursuing a course of action that's going to be actively blocked later on.