Formatting has nothing to do with sector size. Formatting sets the “Allocation Units”.
On an old hard drive with 512-byte sectors, if you formatted with 4096-byte Allocation Units, then each Allocation Unit of data you write to the disk, occupies 8 physical sectors on the drive.
An Advanced Format drive uses 4096-byte physical sectors to store the data… therefore the Allocation Unit has to be a multiple of 4096 (usually 4096 itself is used).
The “alignment” is basically to do with partitioning… you want partitions to start and end on full sectors, but since older OSes seem to think sectors are ONLY 512 bytes, you can create problems. The alignment program slightly adjusts the start and end of each partition, to make sure that it starts and ends on a boundary, instead of in the middle of a physical sector.
Once a drive is aligned, you wouldn’t have to re-align it if you moved it to a different computer, but you would have to re-align it if you formatted it or adjusted the size of the partitions.
Drives above 2TB don’t really “need” larger sectors, per se (although it can help with maintaining data integrity as densities increase)… it’s mostly just that it’s more efficient to use larger physical sectors, regardless of drive size, so more and more drives are being produced with 4096-byte physical sectors. The 2TB issue is a separate matter, as to whether the partition table is MBR or GPT… MBR only has room for 2TB worth of entries… if you try formatting a larger drive with MBR, it will be truncated down to 2TB in size.