I rather liked OTRS. Very easy to set up on Linux and then I rarely ever had to touch it except upgrades, which were trivial. We just used the free edition. I believe it has a CMDB but we didn't use that. Also has an ITSM component if you're into that sort of thing. I found it particularly easy to integrate with email and to have web services that allow for custom forms on a pretty website. Also I found its Perl code to be rather easy to modify when it did something I didn't like. We've been forced into using TechExcel in IT now, which is a big bloated pile of garbage, but the Financial Aid office actually still uses a separate instance of OTRS as a CRM solution to track issues with students' money - again we sold them on it because it was free and not too difficult to maintain and we could hide all the features they didn't need.
We use OTRS here in our school district. http://www.otrs.com/homepage/software/otrs-help-desk-features/
We implemented it roughly 2 years ago, and it has been fantastic for our department of 4 techs and roughly 600 users. It allows for Pending Completions, Users to create an account and see their ticket and the status or send a ticket directly from email. I would highly recommend!
We use OTRS and it can do what you are asking. http://www.otrs.com/en/
The reporting on OTRS is not perfect, so I sometimes create my own reports by going into the MySQL DB and writing my own queries.
Another popular one is RT. http://bestpractical.com/rt/
Would OTRS be up your alley?
It's completely free and open source, which is always a good thing for small startups. The feature list is pretty extensive, too: http://www.otrs.com/en/products/help-desk/features/
There's even an iPhone app.
Why not do your own ticket system and then you can generate reports once a week to show what you've done? Maybe he will ask where you got the report, you show him the system, it's adopted company or department wide, you're the new 5 minute hero.
I'd suggest setting up Virtualbox on your PC, build a small 10GB vHDD 512MB vRAM Debian VM, then install some free ticket system like OTRS or os:Ticket.
OTRS will be able to do what you want by using its PostMaster filters.
More complex definitions can be done in the configuration file.
Even more complex stuff could be written as a module in Kernel/System/PostMaster/Filter/.
I'd suggest OTRS Help Desk. I know from experience it can handle most of the requirements you listed, and it seems that it can handle the others; it's pretty damn customizable (almost to a fault, last time I used it, but that was years ago).
I would look at OTRS ITSM--it's open source, free, and is ITIL ready. I've seen it suggested lots of times here.
If you do have a budget and are looking for something decent and comes with support, I'd look at ServiceDesk Plus by ManageEngine (Zoho). We use it at my job along with two of their other products. I'm impressed with the quality, and their software really isn't that expensive.
Tickets - OTRS. New tickets can be created by email. OTRS has some ITSM support: http://www.otrs.com/software/otrsitsm-features/
Project management - Redmine.
iTop looks interesting as well, but I have not used it: http://www.combodo.com/-Features-.html
Someone took good SW pieces and integrated them together: http://www.smartitsm.org/it_stack
My first argument against Google Calendar would be "do you want to store company-internal information on someone else's computer?".
What's up with sites like these not showing the price anywhere? Is this just so they can charge you as much as they think you'll pay?
http://www.otrs.com/solutions/business-solution-best-practice-edition/
I tested few software for the same purpose some years back and Spiceworks (paid) and OTRS (free) had the best features/ease of configuration and use.
Spiceworks is cloud based so you dont need to worry about setting up any infrastructure youself.
OTRS software is free but you will need to setup your own server ( a cheap vps perhaps?) , configure and maintain it. OTRS Group will provide paid training and support if required.
Does the remote-control thing need to be part of the software? Or could it be just plain 'mstsc' which is launched via the software? If yes, then go for OTRS with the OTRS::ITSM package.
Both are OpenSource, but commercial support/development is also available.
Integrating a link for each ConfigItem which launches 'mstsc' for the corresponding client should be a piece of a cake.
Spiceworks is great if you need a whole lot of features wrapped up in one package, and the community is great. If you just need a ticketing system, I would go with something like OTRS