Are all spontaneous reactions exothermic? Are all exothermic reactions spontaneous?

A spontaneous process must have:

#DeltaG < 0#

...and that's it. This can be accomplished in the following ways (among others), where #T# is positive in #"K"#:

#DeltaG = DeltaH - TDeltaS#

where:
#DeltaH < 0# and #DeltaS > 0 -># always spontaneous
#DeltaH < 0# and [#DeltaS < 0#] and [#DeltaS# is small]
#DeltaH > 0# but [#DeltaS > 0#] and [#DeltaS# is large and/or #T# is high]

Keep in mind that this is a general equation with general entropy; entropy doesn't have to be positive if, say, it's #DeltaS_"sys"# in a closed system.

That aside, taking the third case as an example, if #DeltaH = 24 "kJ/mol"# and #DeltaS = 100 "J/mol"*"K"# and #T = 400"K"#:

#DeltaG = 24 "kJ/mol" - (400 "K")(0.1"kJ/mol"*"K")#

#= 24 - 40 "kJ/mol" < 0#

#=> DeltaG < 0#

You've made the reaction spontaneous by doing it at a high temperature, but it's still an endothermic reaction.

Conversely, with:
#DeltaH < 0# and [#DeltaS < 0# and #T# is high] #-> |TDeltaS| > |DeltaH|#

Example:
#DeltaH_"sys" = -24 "kJ/mol"# and #DeltaS_"sys" = -100 "J/mol"*"K"# and #T = 400"K"#

#DeltaG = -24 "kJ/mol" - (400 "K")(-0.1"kJ/mol"*"K")#

#= -24 + 40 "kJ/mol" > 0#

#=> DeltaG > 0#

and the reaction is exothermic but nonspontaneous.

Therefore, not all spontaneous reactions are also exothermic. Furthermore, not all exothermic reactions are spontaneous.