How do we determine a fair value of forward implied volatility (FVA strike) at inception? Also, I'd be interested in hearing how we can replicate such position? Straddles maybe? Or should we use strangles as they are less sensitive to spot?

I assume we are talking rates FVA. You can hedge part of FVA through a vol triangle (swaptions) or through swaption vs midcurve. Part is because FVA has exposure to forward vol, while those hedges are spot vol sensitive, but it's the best you can do in vanilla space. Otherwise there is not much in the market that has fwd vol exposure - you could offset it with Berms and Vol bonds but unless it is from some other flows usually dealers just sit on the risk. Investors obvisouly take the specifical view on fwd vol.

Price for an FVA should reflect its hedging cost, so here you go.

You need a framework to market make this I think, on the sell side. On the buy side, it's whatever you want. What we all know is that at the forward expiration date, it will be an 'observable' ATM spot vol contract. 

I saw this being bid-offered off of a vanilla swaption and a mid-curve. So as overkill112358 was saying. You would break this down into vanilla swaptions and a correlation exposure. I would not just sit on it. To the extent possible you can hedge the spot vol and you can hedge the correlation using yield curve (CMS) spread options. It's a little hairy to hedge the correlation. 

In any case, you are better off making markets in such products if you have a consistent a framework as possible. I think it is possible to do that, but sure this will be subject to underlying statistical assumptions, which you may or may not like.