Pretty sure a lot of English VAs also sing..... Now, if only I could spoiler tag my seiyuu jabber.
A little unsure what you mean by public image, lol.
We can graps a seiyuu's image, manufactured or not, through radio shows, DVD/BD specials. Sometimes there are tracks on drama CDs where the cast just talks.
Then there's the real "seiyuu idol" part, something Imai Asami is sort of famous for because of her being in the ever popular Idolmaster. I forget if Kuwashima Houko does any real radio shows because she was on Say!You Young earlier this month and her agency did try to make her an idol.... Anyway, Orikasa Fumiko is one of the more popular voices who isn't in the limelight. Kugimiya Rie, as popular as she is, is usually too busy working to do any public events. That's why the Taiwanese went crazy when she went over there.
And there are faceless seiyuu, mostly the ones who only do eroge. These guys don't do any events and stuff, though some are appearing in the light — Sakurai Harumi as Yurippe in Angel Beats! for example.
Singing is a requirement in voice acting school. And it's okay if they suck because Noto Mamiko ohgodstoppingnow.
Dramas.... Depends. Some seiyuu have appeared on dramas at a younger age before really becoming a seiyuu [Kitamura Eri, Chiba Saeko, Horie Yui], some are multi-talented [Yuuki Aoi, Miyano Mamoru, Tomatsu Haruka?] and some have done theatre stuff. Sawashiro Miyuki is the only real name that I can think right now, but she was in this play thing with Mizuki Nana, Suzumura Ken'ichi, Sugita Tomokazu and Konishi Katsuyuki with Wakamoto Norio being the narrator. And Namikawa Daisuke is making some weird movie with a bunch of seiyuu in it and Takahashi Chiaki is hilarious in the clip I've seen.
.... Totally forgot that there are ex-models/singers/actors that sort of dived into the seiyuu business — Fukui Yukari, Kobayashi Yu, Takabe Ai (ry
tl;dr
Voice acting is srs bsns in Japan. Not sure how voicing in a video game gets paid because anime is by the episode and eroge is by the word, but.... :V