No, Talley rings will not need to be lapped. While it will not damage the ring since they're horizontally split, it will void the warranty. If you receive a pair of rings that look like they need lapped, please contact Talley directly so we can get your mounts swapped out.
Medium rings should be fine, item 740000.
I would recommend medium rings. If you have a long action, you'll need item 74X725 or 740725 for a short action.
Medium rings should be fine. It appears that the scope is very short, so you may need to go with a Picatinny rail setup.
I would recommend a medium with extended front, item 74X700.
Yes, medium rings will work fine on that application. You'll need item 740700 or 740700SM (20 MOA).
I recommend using the high 30mm diameter Talley Lightweight Alloy Scope Mounts for that scope and rifle.
A 56mm scope will likely require you to use Extra-High scope rings in order to make clearance for your objective lens. Email us at Techsales@opticsplanet.com and let us know what specific rifle you have, that way we can more accurately recommend a set of rings that will work for your setup.
The exact model of the Talley Lightweight Alloy Scope Mounts you will need will depend on what scope you are mounting to that rifle. The Talley Lightweight Alloy Scope Mounts model numbers 948700, 938700, 958700, 738700, 748700, and 758700 all have the 8-40 screws.
These Talley Lightweight Alloy Scope Mounts will not fit a Browning BAR. However, the Talley Rings located at the link below will fit a Browning BAR.
https://www.opticsplanet.com/talley-lightweight-alloy-scope-mounts-for-benelli-h-r-heym-mauser-surgeon.html?_iv_parent-gun-model=browning-bar&_iv_gun-model=browning-bar-1