Google worked for part of your sample, but not all of it. The following might
not be my final word on the subject, others are encouraged to comment.

Orthographic note:
Neat as it is to use real esperanto letters where they're supported, here in
Fidonet for maximum compatibilty it's old school ASCII only, so in here please
substute ikso formo (X-form), for example `sx' for the `s' with a `^' on top.
The utf-8 or whatever letter comes out as a string of garbage.

fido.ESPOMEN message #178 from Judy Goodman to Rick Ekstrom.
   Entered on 11th July, 2013 at 14:44, 29 lines.
   >
   > Subject: Today's meeting
   > ========================
   >
   > Aside from the sleep situation (was woken too early from a
   > deep, useful sleep by phone ringing -- messed up the sleep
   > cycle), I'm experiencing tummy issues. I think the chopped
   > lamb I cooked for a late dinner may have been bad. So far
   > so good, but I think I'll be in a big rush very soon. If
   > the lamb doesn't get me, the noises in my gut will!
   >
   > For fun, I tossed that paragraph into Google translate to
   > see what they did with the some of the idiomatic language.
   > What do you think?
   >
   > Aparte de la dormo situacio (estis vekita tro frue de
   > profunda, utila dormo telefone sonorado

I probably would have made this "per telefona sonorado", or "(telefono
sonoris)".  Google's adverb ending makes me think that you personally rang
like a telephone does.

> - paneas la dormon ciklo),

make that
 - paneis la dorman ciklon),

that is
  paneas => paneis   to keep your verb past tense,
  dormon => dorman   to make sleep an adjective, and
  ciklo  => ciklon   to make cycle object case.
                      A noun and its adjective must match case.
                     Funny google put -N on Dormo, but not on on Ciklo.
                     Really, how often do two nouns come together and
                     neither becomes an adjective?

> mi spertas Tummy temoj. Mi kredas ke la pikita
> ŝafidon mi kuiris por malfrue vespermanĝi eble
> estis malbona. Ĝis nun tiel bona, sed mi pensas ke mi
> estos en granda hasto tre frue. Se la ŝafido ne
> ricevas min, la bruoj en mia gut volo!
>
> My favorite thing is to then take the translation and
> retranslate it back into English:
>
> Aside from the sleep situation (he was awakened too early
> from a deep, beneficial sleep the phone ringing - messed
> up the sleep cycle), I experienced Tummy issues. I think
> the minced lamb I cooked for a late dinner may have been
> wrong. So far so good, but I think I will be in a great
> hurry soon. If the lamb does not get me, the noises in my
> gut will!
>
> Do you have any clue why "tummy" needs to be capitalized
> in Esperanto?

No idea, esperanto uses fewer caps than does english. Anyway, it should be
"ventro" (belly, abdomen).   Or here,
"ventran" or "ventrajn", since it's object case and is modifying `issues'. 
Probably the singular, more common in e-o, rather than the idiomatic plural
common in English.

"Temo" I most often see for theme, or topic, but I think it works here for
issue, or "matters", or concerns.

> Is there an E. word for hamburger?

Bovburgo, but the bovo part means beef.  I'm not sure whether sxafidburgo
would work.  You could try it.

> "Minced" seems such an odd choice in the original context.

There are probably whole clubs full of esperanto cooking enthusiasts, with
lots of detailed specialized words -- I haven't gotten into it that deeply. 
Yet. How finely chopped was your lamb?
Think on these:

  cutlet   -    kotleto
  chop     -    haki
  chopped  -    hakita
  piki     -    prick

If it's shaped like a patty,
   platbulo  (flat- ball, or lump).
   Bulo is not to be confused with pilko, ball as in sports.

Perhaps `hakita sxafidajxo" would work, but don't take that to the bank just
yet, find somebody who actually knows the fine points of esperanto cooking
words.  Even if I'm close, it might need to be made smaller, maybe something
like `hakiteta'.

> Do I remember correctly that E. is supposed to be a
> balanced, unsexist language?

It's not at all bad for something from the 1800s.  Remember that most natural
languages are much older, and far worse. And to start over yet again from
scratch, would mean you might not be around to see yours get adopted, if it
ever is.

>  Or is it Google that jumped to choose the pronoun in
> without any evidence to back it up?

I didn't  notice any feminine particles there.  Just where are you talking
about?

> Is there no gerund form in E.?

Remind me what a gerund is, using actual examples of speech. I concentrated
more on acquiring a smooth, natural feel for how to do and say things, putting
off at least some formal labels for later (like evidently now).  I've known
people who can recite precise technical descriptions in tremendous detail, but
are almost paralyzed when trying to actually use what they're describing, so
they turn to me for that. My goals were things like to be able to read native
literature, and converse with natives, rather than just discuss those
activities with other scholars, all in english.

I think I made the right choice in personal upgrade path.

--- BBBS/Li6 v4.10 Dada-1
 * Origin: Prism bbs (1:261/38)