1. Is there some sort of a car charger for the battery? I know I can find an answer but maybe someone will have a good suggestion that I might overlook.
Just get a second battery (maybe two)... LP-E6 are rated to 1100 shots with "normal" usage. I use a few power saving tricks and easily get 1200+ per charge. You'll probably need more memory cards, too, if you plan to shoot more than one battery can handle in a day's shooting (Note: I don't use big cards, don't want to put all my eggs in one basket. I have fourteen 8GB cards and a couple 16GB for use in two 7Ds and a 5DII... 8GB is good for about 275 RAW files.)
With all my Canon I've always used vertical grips, which double the number of batteries to two. With earlier Canon digital cameras that used BP511/511A, I used to carry two backup batteries for every battery in-camera. I often had to change out batteries later in a heavy day of shooting. Now using more recent models that use the LP-E6 (the same batt as your 60D, though the cameras are 7D and 5DII in my case) I only carry one backup per battery in-camera. I rarely need to swap in the course of a shooting day, and sometimes shoot upwards of 5000 images a day with a pair of 7Ds (roughly 2500 images apiece, with two batteries in each).
Now, I have my cameras set to sleep quickly (they wake up instantly so this isn't a problem). I also don't use the built-in flash, which gobbles up battery power. Another significant power saving trick is turning off automatic review of every image. I do call up images and their histograms to check them every so often, but not every one. Live View shooting, Mirror Lockup and long exposures all also consume a lot of power. Shooting video does, too. Any time the mirror is held flipped up and the shutter remains open, both are drawing power continuously. On the other hand, metering, Auto Focus and Image Stabilization do not use a lot of power.
Maybe. Depending upon the flash, it might be usable manually, or it may be possible to use with the old style thyristor auto mode (where the flash's built in metering shuts it off when there is adequate exposure).
If the flash is something like an old Vivitar or Sunpak with a single contact, that's universal and not dedicated. I'd think twice about using any of the Canon EZ (early EOS) or xxxA or xxxT (for FD/FL mount cameras). The EZ are partially usable, but limited in usefulness on today's EOS. The xxxA and xxxT Speedlites of the 60s, 70s and 80s have a completely different form of dedication and might cause issues on a modern camera. I don't think I'd even mount on in the hot shoe, just as a precaution.
There is far more automation and ease of use with modern flashes, designed to work with your camera. I'd suggest you consider a 430 EX or 580 EX. Heck, even the simple little 270 EX will often do a better job a lot more easily than those old flashes. There are also some cheaper third party flashes that are pretty good, certainly will be much better than the older flashes. You don't need to spend a lot to get the conveniences of modern ETTL, etc.
There are both wired and wireless remote releases available for your camera. They also can simply be a remote shutter release, or have more features such as timers and time lapse/multi-exposure capabilities. Shop around. Canon offers them, but many others also offer them at lower prices.
Eventually you will need to clean the sensor of your camera. Go to www.cleaningdigitalcameras.com and start reading, decide if you want to do the deeper cleanings yourself or not. If not, at least get a Rocket Blower to puff away loose dust. If you do want to learn to do full cleanings, you'll need more stuff (the first cleaning of any camera must be a "wet cleaning" to remove oily residues.... such as the "Copperhill Method". Subsequent cleanings might be possible with dry methods.)
Lens cleaning is the same as with your film cameras.
I suggest getting the matched lens hood for your lens. It protects the lens and can help your images, cannot do any harm. If the Canon hood seems pricey, search for third party clones that are cheaper.
I just use plastic bags, gaffer tape and rubber bands. I also have $3 plastic ponchos in my camera bags and car, to keep both me and the gear dry.
There are fancier and more expensive "rain coats" for camera and lens, but I've never found them necessary.
You won't damage your camera trying out those lenses, so long as they are EF mount. The Canon lenses almost certainly will work. AFAIK, there are no compatibility issues among EF lenses and EOS cameras, throughout the 20+ years that they've been made.
The Sigma lenses, on the other hand, aren't guaranteed. Some older Sigma are incompatible with newer Canon cameras. As soon as you try to focus, the camera might lock up with an Err code displayed on the LCD. Just remove the lens, turn the camera off and back on... maybe you'll need to pull the battery and reinstall it... no harm will be done, but the lens won't be usable. I have an older Sigma 28-75mm that works fine on EOS-3 (film) and 10D (digital) but won't work with more recent 50D, 7D, 5DII cameras.
If you send them in for service, Sigma will fix some of the higher end lenses, updating them to work on newer Canon. If you have a Sigma 70-200/2.8 or 300/2.8 or 500/4.5, it might be possible to have it updated. But a cheapie like my 28-75 isn't even worth sending in (I paid $7 US for it).
You sound as if you have some photography experience, so may or may not find the book "Understanding Exposure" by Bryan Peterson helpful. But, hey, it even taught this old dog some new tricks, so you might enjoy it or find it a useful refresher.
I would recommend buyingone or two of the guide books specifically for 60D, to help you get up to speed with the camera as quickly and easily as possible. I don't have 60D, but have bought guide books by Charlotte Lowrie, Michael Guncheon and David Busch, and feel comfortable recommending all of them.
You mention some older EOS film cameras, but if you are new to the Canon AF systems of today, suggest watching this YouTube video, plus the other two in the series (each is about 1/2 hour). These cover the basics and gets into a lot helpful details about using the systems on various cameras.
If you haven't done much digital photography, post production is another big consideration. In effect, you are now your own film processer and print tech. No more sending stuff to a lab and waiting to get the results. There's a learning curve to this, too. You might need more computer storage, RAM, a better (graphics quality) display monitor, monitor/computer/printer calibration, and will want to develop an efficient workflow. Canon provides some pretty good software with your camera to start with. Look for some books about the post-production process and start learning! I recommend The DAM Book - Digital Asset Management for Photographers as a good starting point setting up an efficient workflow. It's been revised a few times since I bought my copy, used to be pretty specific about softwares and methodology, although there are some equally good alternatives. But, the overall intent, princibles and recommendations are very good and valid, even if one adapts the ideas a bit for their own use.
You may want to hold off buying additional lenses for a while. For one thing, you probably already know that as an old film shooter with your first DSLR, you have some preconceived notions about focal lengths that are wrong now, considering the smaller sensor format and so-called 1.6X APS-C "crop factor". The 18-135mm you got with the camera is a pretty versatile lens covering moderately wide (29mm equiv. on your film cameras) to moderate telephoto (216mm equiv. on film). 250mm is a pretty powerful tele (400mm equiv. on film). 300mm,m 400mm and 500mm are super telephoto (480mm, 640mm and 800mm equiv. on film, respectively).
You don't need UV filters the way you might have with film... Digital cameras filter out UV all on their own, while some films were overly sensitive to it and often required filtration. A lot of people use UV filters today for "protection", but I think it's kind of silly to expect a thin piece of glass to provide much practical protection... So I generally use lens hoods instead. They offer more realistic protection and can't get in the way of good images the way some filters can.
The most useful filter today, IMO, is the Circular Polarizer. It's one filter that's nearly impossible to replicate well in post-production software. Most other filters are extraneous now. For example, set custom white balance now, instead of using color correcting filters.
Have fun with your new camera!