This pronunciation sounds a bit off to me - shouldn't it be "JEH-chee?" Sounds like she's saying "JEE-chee"
It does sound like that in the v recording. You'll sometimes hear such 'tense' pronunciations of e from Poles, but there's no reason to learn or imitate it.
Do you mean 'having' as in 'having lunch'? Because that's the same sentence in Polish.
Polish conjugates verbs according to the grammatical gender. So you have:
(ja) jem = I eat/I am eating
(ty) jesz = you eat/you are eating (singular 'you')
(on/ona/ono) je = he/she/it eats / he/she/it is eating
(my) jemy = we eat/we are eating
(wy) jecie = you eat/you are eating (plural 'you')
(oni/one) jedzą = they eat/they are eating