There are none. The only internationally recognized countries that start with A at all are as follows:
Albania
Algeria
Andorra
Angola
Antigua and Barbuda
Armenia
Australia
Austria
Now this is as they are spelled in the English Language and in the Latin alphabet. Other Western Languages such as French might have countries that start with an A and end in an E. Languages that do not have a Latin alphabet might have words for countries that start with an A vowel sound and end in the E vowel sound.
Can we "START" ?
zaniestzealotzeitgeistzestzestiestzibetzigguratzionistzippiestzitzitheristzithernistzoograftzoologistzoophilistzootomistzymologist
oozily
no it isn
yion
BelizeCape VerdeChileCote d'IvorieFranceGreeceMozambiqueSao Tome and PrincipeSierra LeoneSingaporeSurinameTimor LesteUkraineZimbabweThese are the only fourteen countries ending in 'e'.
Some eight letter words starting with F and ending in E are:favoritefeminineflatlineflexibleforciblefracturefugitive
elfengulf
LaurieLizzieLoraineLyle
zone
eve,ese(spanish),eveedgeeweenzymeexposeexpertiseenvelopeemoteeludeevadeelectrodeencaseendureevadeevaporateevokeestimateevaluateestateextremeexploreexchangeexpireexampleexfoliateexterminateeliminateeconomizeequalizeexhaleexhume
Afghanistan and Azerbaijan.
yes
There are no four letter words in English starting with e and ending in q.
Ending
eventful, that's about it
Teacake