You're trying to analyze a fragment, which will certainly show inconsistency since the rule that affects it operates at word level.
The e in the middle of appeler is an example of what's called an "e caduc" (also known as muet, facultatif, or instable). The context determines whether it's pronounced or not. When it occurs in the middle of the word, as with appeler, its dropping out is an example of syncope.
Essentially, this vowel was once or is nominally present in the word, but depending on the stress pattern, which depends on the conjugation, it can be heavily reduced. Reduced vowels tend to drop out; the phenomenon is so natural cross-linguistically that it might have been done in Latin, too.
Dropping it is mostly systematic in French, so I would hesitate to call it a mistake. It would be odd to say « Je vais à-pelle-ai mon frère » instead of « Je vais appler mon frère » !
The stress that affects whether this e is reduced is actually not a special case but derives from the rule that the stress falls on the last syllable in French (at least when isolating grammar from usage). In the conjugations that have a syllable after the e caduc, said syllable gets the stress. Therefore, the e is reduced and liable to be dropped:
/ap(ə)l-/ : appelons, appelez, appelais, appelait, appelions,1 appeliez, appelaient
/apɛl/ : appelle, appelles, appellent
The same rule covers lever, renouveler, acheter, amener, and others.
1 In appelions and appeliez, the e can sometimes be heard, as in the conjugation table you quoted. My interpretation of that data is that the onset cluster /.plj/ is pretty unlikely. Phonologically I'm not sure whether the e caduc surfaces as a schwa, as in /pə.lj/, or the /p/ is moved to the coda and released, as in /p.lj/ — but phonetically the result would be about the same.