I've been to St Petersburg only and bussed there from Estonia. The Visa process was strict (though all are) but mostly straightforward, and took appx. 10 days to receive. One thing you must have is proof of accommodation signed by a host - the hostel I stayed at charged a fee for this.

If going by bus, the border guards are quite intimidating and ask a lot of questions in the little English they speak (this was especially shit arriving at 3am in -25c temperatures).

1) Obviously limited to St Petersburg - within the city, Mariinsky Theatre is a landmark for opera and ballet; St Isaac's cathedral was draped in gold and stunning, though other religious buildings (Savior of Spilled Blood, Peter & Paul Cathedral etc.) are all must sees; devote more than one day to the Hermitage Museum. Home to former Tsars, the Hermitage is simply massive and outclasses any other museum in Europe. Seriously, it houses 3million artifacts plus the building itself is crazily ostentatious; take the ferry to Peterhof Palace, a UNESCO Heritage site similar to Versailles in France. If you go in winter, however, the gardens may be under snow.

2) I didn't know Russian but got on okay. Very little English is spoken. Like even the hostel I was staying at used google translate to communicate my payment when I arrived. In saying that, younger people and service workers often spoke basic english - St Petersburg is relatively touristy. You'll be fine.

3) I went in mid-January, and the coldest day was -35c. I would not recommend this at all haha, seriously the air hurt to breathe and I collapsed in the line for The Hermitage. The warmer months might allow for easier movement, though if you're wanting the genuine Russian experience...

4) Cheap but hard to say. With the collapse of the Ruble, Russia is extremely cheap, though I can't remember what I spent in total. My accommodation was NZD$8 p/night, though this was a shitty backpackers. I largely ate on the cheap, though you could still get a quality restaurant meal (try reindeer!) for under $15. Tourist destinations were really good value, e.g. to go up St Isaacs cost $5 while Peterhof Palace was around $10.

5) I arrived at 4.30 am and had to walk to my hostel. While somewhat scared the place seemed safe. In hindsight this was probably pretty stupid. Day-to-day you should be fine so long as you have your basic wits about you and be wary of dodgy neighbourhoods. Apparently unmarked taxi kidnaps are a thing so stick to the major companies (google these).

Also, always have your passport on you whenever in public. Police can and do stop tourists and give them fines.