Not unless it's in a poem or play that's a century or more old! In modern English, the only past tense forms in which 'ed' is pronounced as a separate syllable are those where the base form of the verb ends in a 't' or a 'd'. For example 'wanted' or 'needed'. The simple reason for this is that it's impossible to say these words any other way. We have to pronounce 'needed' as /ni:did/. With all other verbs, the past tense suffix /d/ or /t/ is voiced immediately after the final consonant of the verb. This includes those words ending in a 'th' sound. Some non-native speakers may find it difficult or unnatural to pronounce the consonant cluster 'thd' /ðd/, but it native speakers don't. The pronunciation of 'bathed' is /beɪðd/. Other similar past tense forms pronounced in this way are 'soothed' and 'loathed'.

No, you don't pronounce the "e".