Hi Adriana, welcome to the forums.
I'm an ex-HMC shooter (we had 4) and now have 5 (soon to be 6) Canon DSLRs, including the 60D. We now shoot weddings exclusively on DSLRs.
As you are well aware by now, the LCD on the HMC it not the sharpest around. In fact, it's positively soft. You cannot (let me repeat this, you CANNOT) judge focus using the LCD without also using the focus assist function that zooms the LCD in to 100%. Even then, the LCD is still soft, so I always found myself manually zooming the camera in (we almost never had servo zoom engaged), then use the focus assist (so now we're zoomed all the way in to the subject and have focus assist on too), focus, turn focus assist off then zoom out. All in all, this takes about 7 seconds, but can be done without turning record off (so at least you can maintain continuous audio).
I never found the HMC auto focus to be reliable enough, either to achieve accurate focus quickly, or to focus on what I wanted it to focus on. If you like your subject smack in the middle of the frame then it's OK, but if you are trying to frame your subject properly then manual focus is the only solution.
The ONLY time we ever used auto focus was for the Bride & Groom walking directly towards us or away from us, e.g. Processional and Recessional. Period. End of Story. I know others use it a lot more, but then they complain about missed focus and all I can say is 'told you so" (sorry - didn't mean it in a bad way!).
Now, the 60D is a completely different animal. The LCD is relatively sharp, but of course the 60d has no auto focus built in while recording so you absolutely have to stop, refocus then restart again.
The best way to focus on the 60D is to use the zoom button (DON'T zoom the lens for focus because SLR lenses don't maintain the same focus throughout their range). So, the zoom button is the top right button on the back with the little magnifier and the + sign. (Sorry if you already know this, but others may be reading this who don't). Press this once and it zooms in to 5x, press again and it zooms to 10x. Press again and it goes back to full view. So, at 10x you can generally achieve accurate focus, at least in reasonable light

If you don't want to reframe in order to see the subject on screen while at 10x then use the cursor (the inner dial around the 'set' button) to move which part of the subject is being magnified.
AF on the 60D is achieved by moving the camera until the red focus point is over the subject then pressing and holding the AF-ON button on the back. Simply pressing it won't focus, you need to hold it until the camera has actually focussed. Once focused, reframe and hit record.
Of course, the 60D has the same issues as any camera. AF can be confused by objects close to your subject, including backgrounds. You also have the issue that fast lenses (e.g. f1.4) can have fairly shallow depth of field, so if you are even marginally out, you lost. The same goes for long lenses. We generally have one of the cameras at the back of the isle with the 70-200 f2.8 on it and at 200mm f2.8, the depth of field (the in focus bit) is only 0.79 feet. So, you can have the couple in focus OR the officiant, but not both!
If you have slower lenses (e.g. f4.5-5.6) then focusing is always going to be much harder because they are letting so much less light in.
In terms of tripod vs hand hold, while it's possible (with practice) to hand hold the HMC and get relatively steady footage, we shot 99.9999999% of the time on a tripod. I wouldn't even think about shooting the 60D hand held because the CMOS rolling shutter is going to pretty much destroy the shot. Maybe you can do it with a shoulder brace, or a Glidecam if you have the time to set it up, but the tripod will always provide a more stable shot, something that is critical with CMOS sensors.
I'm also interested in how you are finding the 60D and HMCs match in post production. I could never get them to look close enough to be happy. The DSLR footage just stood out head and shoulders above the rest. Maybe you have a secret formula you'd care to share?