Can I run my old fridge off the battery with inverter?

Extra insulation is good as long as it doesn't interfere with the heat dissipation, as you said. The good thing about the old fridges is that all the heat comes out the back. New electric fridges dissipate heat through the metal box sides and top, so extra insulation would hurt performance (unless you tear the thing apart, get rid of the metal box and re-position the cooling tubes outside the new insulation). I'd try and find installation instructions for that particular fridge though, just in case it requires air space around the sides and top for some unknown reason.

Absorption fridges are pretty quiet (especially with no burner to roar), and the heat might be on for some amount of time before you hear any gurgles or anything (making the timing late), so the thing to do might be to leave the meter clamp on it and watch for the amps to jump. I don't think it's necessary to watch it for 24 hrs, though. I'd do it for the hottest hour or two of the afternoon, when it will work its hardest, and use that maximum duty cycle as a constant.

If, when doing all the usage calculations for your battery bank size, in matters of uncertain exactness, you always "round up" or err on the side of more usage rather than an optimistic minimizing (or an attempt to be exact which can't foresee everything), you will build an extra margin of capacity into your system, which could come in handy for unexpected electrical needs. In this case, timing the fridge during the hottest part of the day and using that figure as a constant 24-hr cycle will help keep you running if you go somewhere where it gets hotter than it is now, or travel with a mostly-empty fridge, that makes it work harder.

In the case of say, a coffee maker, you might time it and see that it takes 10 minutes to make a pot and use that as your time quantity when you calculate your battery needs. But it would be prudent to add some time to allow for you not always pouring it immediately and turning it off immediately. You don't want to have to forgo using your fan in the afternoon just because you were in the bathroom or chasing the dog when the coffee was ready.

You might intend to shut off the lights and go to sleep at 10pm every night, but again, it's prudent to allow extra capacity for the nights that you fall asleep with the lights on.

If you have a video camera or webcam, you could just set it up so it's aimed at the meter and a clock, and then instead of staring at the thing, you could fast-forward the video to find all the on/off points...

I came across an interesting article on deep cell RV batteries, and how they should be charged properly and what is a "true" deep cell battery. I don't live off grid, so I don't know that I need to be as diligent as the author describes, but there is some interesting info in his discussion about how to charge your batteries correctly,for anyone who is interested in this kind of thing...

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