Remove RootContext — adopt handler constructor DI

## Background

Manager API handlers currently depend on `RootContext`, a god object with ~50 fields. The goal is to eliminate this dependency by setting up all dependencies via `DependencyComposer` before server setup, then injecting only the required dependencies into each Handler constructor.

## Target State

```python
resources = await stack.enter_composer(ManagerDependencyComposer(), input)
session_handler = SessionHandler(
processors=resources.processing.processors,
registry=resources.agents.registry,
)
```

## Tracks

- **Track A (Composer Development)**: 7 independent Composer modules
- **Track B (Handler Conversion)**: function-based → class-based constructor DI
- **Track C (Integration)**: Composer integration + create_app() wiring + RootContext removal

## Phase Diagram

```
Phase 1 (parallel, 9 issues) Phase 2 (integration)
┌─ Plugins ────┐
├─ Messaging ──┤
├─ Domain ─────┤
├─ Agents ─────┼──→ Composer Integration ──┐
├─ Orchestr. ──┤ ├──→ Wire create_app() + Remove RootContext
├─ Processing ─┤ │
├─ System ─────┘ │
├─ Handlers 1 ────────────────────────────┤
└─ Handlers 2 ────────────────────────────┘
```

JIRA Issue: BA-4395

Start by mapping the Manager API handlers, RootContext, DependencyComposer, ManagerDependencyComposer, and create_app() wiring described in the issue. Review the three tracks and their dependencies before choosing a focused conversion task. Done means handlers receive only required constructor dependencies, create_app() uses the composer, and RootContext is removed.

Written by the indexing model from the issue text.