Expanding the memory via sd card will it increase the entire phone memory for apps or just storage?
It only gets you more space for video, audio and stuff like PDF and office documents. You can't install apps to the SD card. The one exception is Nokia Drive, which allows you to install offline maps to the SD card (which is great as these are often quite large, > 1GB)
920 is in almost all aspects is the better choice.
because 920 is better than 820 in almost all aspects
I disagree with such statements. IMHO the first consideration should be the internal hardware (1.5GHz CPU and 1GB RAM, etc), and for both the L820 and L920, these are absolutely identical.
Based on your choices, which includes only the older 1GB models, I'll assume you are a price conscious mobile gamer. If that doesn't describe you, then you need to rethink your selection.
The main reasons to get L820 over the L920
were expandable storage, the slightly smaller and lighter build, preference for AMOLED screens, and price. If you are deciding based on any of the last three criteria, then Nokia now has much better options. The only remaining reason to get a L820 today, is if you want the high-end specs (1.5GHz CPU and 1GB RAM) and expandable storage.
The only reason to consider a L920 today is price. It is cheaper than the L925 and the L928 which are the better options. The main thing the 920 has going for it over the L820 is the higher resolution display (which as mentioned isn't necessarily better if you prefer AMOLED) and the camera. If you don't know whether you prefer AMOLED over LED, then you need to look at both so you can decide for yourself which is better for you.
Personally, I'd prefer the L720 over the L820 (looks better, feels better, lasts forever on a charge, also has expandable storage and is cheaper). I'd also prefer the L925 over the L920 (because I much prefer AMOLED and I also appreciate the smaller/lighter build).
Edit: At similar price points, if you have no display technology preference, you don't need expandable storage, and if you don't mind the heft, definitely get the L920.