No, you cannot use 8-bit or 16-bit registers in addressing calculations in 64-bit mode, nor can you use 8-bit registers in 32-bit mode. You can use 16-bit registers in 32-bit mode, and 32-bit registers in 64-bit mode, via use of the 0x67 address size prefix byte.
(But using a narrower register makes the whole address narrow, not a 16-bit index relative to a 32-bit array address. Any registers need to be the same width as the address, which you normally want to match the mode you’re in, unless you have stuff in the low 16 or low 32 bits of address space.)
This table summarizes well the various options for operand and address sizes. The general pattern is that the default address size is the same as the current mode (i.e., 32-bits in 32-bit mode, 64-bits in 64-bit mode)1, and then if the 0x67 prefix is included, the address size is changed to half the usual size (i.e., 16-bits in 32-bit mode, 32-bits in 64-bit mode).
Here’s an excerpt of the full table linked above showing 64-bit long-mode behavior only, for various values of the REX.W, 0x66 operand and 0x67 address size prefixes:
| REX.W | 0x66 prefix (operand) | 0x67 prefix (address) | Operand size (footnote 2) | Address size |
|---|---|---|---|---|
| 0 | No | No | 32-bit | 64-bit |
| 0 | No | Yes | 32-bit | 32-bit |
| 0 | Yes | No | 16-bit | 64-bit |
| 0 | Yes | Yes | 16-bit | 32-bit |
| 1 | Ignored | No | 64-bit | 64-bit |
| 1 | ignored | Yes | 64-bit | 32-bit |
1 That might seem obvious, but it’s the opposite to the way operand sizes work in 64-bit mode: most default to 32-bits, even in 64-bit mode, and a REX prefix is needed to promote them to 64-bits.
2 Some instructions default to 64-bit operand size without any REX prefix, notably push, pop, call and conditional jumps, and as Peter points out below, this leads to the odd situation where at least some of these instructions (push and pop included) can’t be encoded to use 32-bit operands, but can use 16-bit operands (with the 0x66 prefix).