Sorry to hear your camrea was stolen...
Coming from 20D, you'll see 1.5 to 2 stops higher usable ISO with any of the 18MP crop sensor models right now, including the 60D. You'll just have to scale back on any pixel peeping you're in the habit of doing.... Looking at a 20D image at 100% on your computer monitor is like making a 3 foot wide print from it. 100% with the much higher resolution images from the 18MP cameras is like making a 4-1/2 foot wide print and viewing it from the same distance. Going from 12 bit to 14 bit also means better image quality, along with much larger files that might need updated software to handle them. (Depends upon what s'ware you have... when I went from 30D to 50D I had to update Photoshop as well as my operating system... ended up just buying a new computer!)
You could get a used 50D, a very nice camera that's quite similar in size, shape, layout and controls to your 20D (more-so than the 60D, which uses a bit different controls and layout.) Compared with the 15MP 50D you'd see about 1 stop higher usable ISO for that low light shooting.
You can't avoid video on the current models. They all have it. They all also have Live View. Even the 50D has Live View (but it doesn't have video). The 60D's articulated LCD screen can be handy for low or high angle shots, along with video. It also can be turned face-in, to protect it while shooting or in storage. 50D and 7D don't have an articulated screen.
60D's AF system is very similar to 40D and 50D, a nice step up from 20D/30D's but an easy change to make since it still feels very familiar. The 7D has essentially the same imaging and high ISO capabilities as the 60D, but is bigger/heavier, more durable and fitted out with a much fancier AF system. But many of the 7D's bells and whistles are related to AI Servo shooting with moving subjects, sports/action. It would probably be overkill for your purposes.
If you don't have any or few crop-specific lenses remaining, this may be a good opportunity to change to full frame if interested in doing so. The 6D might be worth a look. This would give you around 2.5-3 stop higher usable high ISO for low light RAW image files... even more if shooting JPEGs. It would also give better low light autofocus performance (Canon rates it to -3 EV, which is approximately moonlight). The 6D is more compact and lighter than any of the other FF Canon. It uses a control layout similar to the 60D's.... a bit different from what you are accustomed to. Of course, a 6D is a lot more expensive than 60D.
Another full frame possibility, used 5D Mark II might serve very well for most of your purposes, too, giving similar high ISO performance with RAW images, but JPEGs and not quite as usable AF in low light (-1 EV). With 5DII, you'd likely want to stick with Center AF Point only when shooting moving subjects like dogs. It's not the best camera for AI Servo/sports/action... but can manage if expectations aren't too high! 5DII controls would feel very familiar to you. It's AF looks and is similar to 20D's, too. There are some good deals on used 5D Mark II right now.
I'm going to assume 5D Mark III is out of the question due to price.
With these 18 to 22MP cameras you might need bigger/faster memory cards. 60D and 6D both use SD memory instead of Compact Flash, anyway... so you may need to budget for this as well. It's possible too that you'll find you want to do some computer upgrades such as faster processor and/or more RAM and/or additional hard drive storage space to handle the much larger files you'll be dealing with.
Hope this helps!