Yes, of course! Anopheles and O'nyong-nyong virus is one example, Anopheles and Jamestown Canyon virus is another. With Culex ... so many.  West Nile virus, St. Louis encephalitis virus, Murray Valley encephalitis virus, etc.

Yes, a surmounting work have shown the evidence. Here are some ; http://www.ncbi.nlm.nih.gov/pubmed/20222979

https://www.clarku.edu/~tlivdahl/page19/kasia/vectordiseases.html

All types of mosquitoes( including anopheles and culex) are known to transmit rift valley virus and culex is a vector for japanese encephalitis virus

Yes of course, there are plenty of references available. Mosquitoes are haemophagus, so all species should be considered as vector  and / or mechanical transmitter of disease pathogen There is a certain percentage of mosquitoes in each species, which have good genetic relation with pathogen / virus - genetic compatibility. However, relatively few species are known as vector  because we want positive result. I give an accidental discovery of Masonia uniformis found  habouring CCHV, which is transmitted by ticks. We were all surprised. Much of it depends also on feeding behaviour of mosquito species and availability of host. 

yes

A word of caution to add to my esteemed colleagues' remarks and to endorse Muhammad's caution: not all susceptible mosquitoes are vectors. Take Rift Valley fever virus for example - many species of Aedes, Anopheles and Culex have been found infected in the field but only a handful of Aedes and Culex species have proven to be capable of transmitting the virus. Infection may be confined to the bloodmeal, to the midgut epithelium and to the salivary gland epithelium, depending on the presence of permissive viral receptors, but if there is no escape of virus into the salivary gland lumen and mixing with saliva, then no transmission can take place.

Referring to Rift Valley fever past epidemics in Egypt, both infected Culex pipiens and Ochlerotatus caspius are able to transmit the virus experimentally.  However, only Culex pipiens has been found naturally infected during these epidemics. Based on epidemiological evidence,  O. caspius was suggested to have acted as a bridge vector between animals and humans in southern Egypt.  Ffindings were published in 1979, 1980, 1987,1994, 1996, 1999.

Fully agree with Alan Kemp...FYI several species of Culex, for example, Culex vishnui, Culex tritaeniorynchys, Culex bitaeniorynchus, Culex gelidus are known vectors of Japanese encephalitis virus

Transmission of myxoma virus and others

1.http://www.parasitesandvectors.com/content/8/1/421

2.http://www.globalhealthaction.net/index.php/gha/article/view/25929

3.http://link.springer.com/article/10.1007%2Fs00436-014-4237-6

Allan,

Thanks for the caution

Numerous Culex species have been confirmed as vectors of a number of different arboviruses.  Depending on the geographic location, West Nile virus is primarily transmitted by Culex tarsalis (western US); Culex pipiens (northern US and much of the world where WNV is found); Culex quinquefasciatus and Culex nigripalpis (southern US); Culex theileri, Culex univittatus, and Culex vansomereni in Africa; etc.  Japanese encephalitis virus is transmitted by a number of mosquitoes in the Culex tritaeniorhynchus complex. St. Louis encephalitis is primarily transmitted by members of the Culex pipiens complex.  Sindbis virus (an alphavirus) is similarly transmitted by various Culex mosquitoes depending on geographic location including Culex pipiens, Culex quinquefasciatus, Culex theileri, and Culex torrentium.  This is just a short list of Culex mosquitoes and viruses associated with them.  There are many more.

In contrast, Anopheles species are rarely implicated as primary vectors of arboviruses.  The primary exception is O'nyong-nyong virus which has been associated with Anopheles gambiae. 

Anyone ever isolated chikungunya from CX.vansomeren...univittatus or any other culex mosquito?

I am not aware of any isolations of chikungunya virus from any Culex mosquito. However, I would not be surprised if someone were able to detect chikungunya virus in a field-collected Culex mosquito during an outbreak. Note, this does NOT mean that the Culex mosquito would be involved in the transmission of this virus. First of all, many people are now using PCR to detect viral RNA, and if a Culex mosquito had fed on a person infected with chikungunya virus, they would likely to be positive for chikungunya viral RNA if sampled. Second, if a single leg of a chikungunya virus-infected Ae. aegypti or Ae. albopictus were to break off and fall into a pool of Culex mosquitoes, that pool would be VERY positive for chikungunya virus by qRT-PCR, with a Ct value of about 24. It would also be positive for infectious virus, with a titer of about 10^4 PFU/ml. In neither of these cases would the Culex mosquitoes actually be infected with chikungunya virus. However, I have seen papers in the literature with other viruses that have claimed infection based on the above. Finally, even if a Culex mosquito could actually be found that contained virus, what does that mean? Could the virus replicate in that mosquito? Could it get to the salivary glands? Could the mosquito transmit it? If not, then, for all purposes, it is a false positive.

A. gambiae is known transmit arbovirus causing O'nyong nyong disease

An. quadrimaculatus can transmit Mayaro virus (MAYV; Alphavirus, Togaviridae), an emerging pathogen endemic in South American countries.