The first issue may be related to fuel supply &/or fuel pressure.
Are hot starts equally as difficult? If not, this may be a faulty fuel accumulator. If so, check for vacuum leaks, and then check ignition, plugs, fuel pumps. Is the main fuel pump (under car, under LHR seat) noisy? If it is, it's either cactus or the in-tank pump is. How old is the main fuel pump filter? If in doubt, replace it. Is the in-tank fuel pump working?
The second issue may indicate you need to replace the engine mounts and the the transmission mount. IIRC (without looking at the book) V6 mounts are different to 4-pot mounts.
Bonus issue - if that missing hose hooks up to a cold air intake, it may be worth replacing; if it hooks up to an exhaust manifold (to provide a form of air-preheating for tundra temps) it's probably not.