Improve `TreeMesh` MPI parallelization

Performance:
- [ ] Avoid allocations during `MPICache` initialization (came up in https://github.com/trixi-framework/Trixi.jl/pull/898#discussion_r737485962)
- [ ] Current way of storing MPI mortars might be problematic in 3D (see https://github.com/trixi-framework/Trixi.jl/pull/898#discussion_r729548386 and https://github.com/trixi-framework/Trixi.jl/pull/898#discussion_r729588551)
- [ ] Current way of storing MPI connectivity might not be efficient enough (for 3D) (see https://github.com/trixi-framework/Trixi.jl/pull/898#discussion_r729580943)
- [ ] MPI communication in load balancing may be iniefficient (see #402)

Other:
- [ ] Remove MPI specific parts from serial code (see e.g. https://github.com/trixi-framework/Trixi.jl/pull/898#discussion_r737465055)
- [ ] Use `uEltype` in MPI buffers (see issue #910)

Start by reviewing the TreeMesh MPI implementation and the MPICache initialization, then read the linked pull-request discussions and issues #402 and #910. The scope spans allocation, 3D mortar and connectivity storage, load-balancing communication, serial-code separation, and MPI buffer element types; done means resolving the relevant checklist items with performance and correctness evidence.

Written by the indexing model from the issue text.