TP has slower max and cruise speeds,
Makes sense, since the propeller tips are exposed and therefore get transonic at high speeds - they lose efficiency and also get loud. Mind you, this is equally applicable to propeller planes with infernal combustion engines.
But TP can be pretty fast yet still efficient. Tu-114 and Tu-95 are somewhat slower than B-52 or B-707, but they have a rather good range. Tu-95 is indeed loud.
And of course a jet can be slow. See Cri-cri twinjet, or Yak-40, or Virgin Global Flyer. But a jet tends to lose efficiency at slow speeds - too much fuel goes to moving the jet rather than the plane. Turboprops are better - if you want a slow turboprop, you just make a bigger propeller with lower rpm and gearbox with a bigger gear ratio. No wonder Yak-40 is the Fuel Destroyer. On the other hand, Global Flyer does achieve a rather impressive range.
lower max and cruise altitudes,
Because of lower ratio between cruise and takeoff TAS, correct?
shorter takeoff and landing requirements,
Because of lower takeoff IAS?
shorter takeoff and landing requirements, possibly better Xwind capability
How do turboprops achieve higher slip angle?
and are generally smaller aircraft.
Generally. On the other hand, compare Tu-114 with Yak-40. Or even ATR-72 with ERJ-135.