I did a mistake. I was using the old generated messages. I generated new messages now, and things changed.
The only message that does not work the right way is HTML + attachment + signature. The "decypted" message and "signature verified" message are missing. But decryption works fine.
Content-Type: multipart/signed; micalg=pgp-sha1;
protocol="application/pgp-signature";
boundary="=_dfa816b4e353745a697c0959358b80aa"
This is an OpenPGP/MIME signed message (RFC 4880 and 3156)
--=_dfa816b4e353745a697c0959358b80aa
Content-Type: multipart/mixed;
boundary="=_6add61e941ee3ae96dc0b235baf64610"
--=_6add61e941ee3ae96dc0b235baf64610
Content-Type: multipart/alternative;
boundary="=_8dded80302f1837e475b3c8bd7a879c8"
--=_8dded80302f1837e475b3c8bd7a879c8
Content-Transfer-Encoding: 7bit
Content-Type: text/plain; charset=US-ASCII
asdasd
asdasd
asdasa
--=_8dded80302f1837e475b3c8bd7a879c8
Content-Transfer-Encoding: quoted-printable
Content-Type: text/html; charset=UTF-8
<html><head><meta http-equiv=3D"Content-Type" content=3D"text/html; charset=
=3DUTF-8" /></head><body style=3D'font-size: 10pt; font-family: Verdana,Gen=
eva,sans-serif'>
<p>asdasd</p>
<p>asdasd</p>
<p>asdasa</p>
</body></html>
--=_8dded80302f1837e475b3c8bd7a879c8--
--=_6add61e941ee3ae96dc0b235baf64610
Content-Transfer-Encoding: base64
Content-Type: text/plain;
name=globdata.ini
Content-Disposition: attachment;
filename=globdata.ini;
size=1110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--=_6add61e941ee3ae96dc0b235baf64610--
--=_dfa816b4e353745a697c0959358b80aa
Content-Type: application/pgp-signature;
name=signature.asc
Content-Disposition: attachment;
filename=signature.asc;
size=819
Content-Description: OpenPGP digital signature
-----BEGIN PGP SIGNATURE-----
Version: GnuPG v1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=2UJ0
-----END PGP SIGNATURE-----
--=_dfa816b4e353745a697c0959358b80aa--
About being able to download inline signature in a plain text email, I don't remember about Enigmail, but for the sake of consistency it would be good to be able to always download it.