[Bugfix] fix bug for mtp (#6514)

### What this PR does / why we need it?
fix(mtp): resolve MTP core bugs and enhance eager mode test cases
1. Resolved critical issues in eager mode MTP core execution logic;
2. Fixed functional bugs in the _update_states_after_model_execute
function;
3. Updated and released test_mtp_qwen3_next.py to validate eager mode
acceptance rate.
### Does this PR introduce _any_ user-facing change?
None
### How was this patch tested?

- vLLM version: v0.15.0
- vLLM main: https://github.com/vllm-project/vllm/commit/v0.15.0

Signed-off-by: Bowen-Leee <caoshankuangren@gmail.com>

tests/e2e/multicard/4-cards/spec_decode/test_mtp_qwen3_next.py

@@ -54,7 +54,6 @@ def test_qwen3_next_mtp_acceptance_tp4(model_name):
                     distributed_executor_backend="mp",
                     disable_log_stats=False,
                     speculative_config={
-                        "cudagraph_mode": "FULL_DECODE_ONLY",
                         "method": "qwen3_next_mtp",
                         "num_speculative_tokens": 3,
                     },