1) As far as I can remember, "Solo" queue actually lets you queue up to a team of two. This may have changed. Unfortunately, Solo Queue is just that - you and some strangers - and if you want an easy strategy to climb the solo queue ladder, there isn't one. You're largely at the whim of the matchmaker, so the best you can hope for is to carry a team, perhaps by playing a ganking jungler such as Rammus, Shaco, or Nocturne.
The current metagame theory is that teams with a jungler > teams without, so by playing a strong jungler (assuming you aren't overestimating your skill) will give you the best odds, as an early game lead can be hard to overcome.
2) Queue dodging in a ranked game no longer reduces your ELO, but it does prevent you from queuing for ranked games for a substantial amount of time. If you want to raise your ELO, you need to play games, and you can't do that if you constantly queue dodge.
It depends. At a certain point, day IP boosts will become worth it. Let's do a little quick math. For simplicity, I will assume you get 100 bonus IP for the IP bonus. Some games are more, some are less, but my experience is that it averages out to ~100.
A 6300 IP champ is 975 RP, making each RP spent ~= 6.46 IP.
4 win IP boost is 290 RP. 1 RP ~= 1.38 IP.
10 win IP boost is 670 RP. 1 RP ~= 1.49 IP.
So you can rule those out. Even with 1 hour long games and 200 IP bonuses (rarely happens), you're still not even close. Now it gets a little more tricky. Number of games that you get the bonus for in a day boost are only limited by the 24 hours in the day. I'll assume 10 games played each day, and each lasting ~ 30 minutes for ~100 IP bonus per game (About 6-7 hours of gaming when you get the IP boost).
1 day is 290 RP. 1 RP ~= 3.49
3 day is 520 RP. 1 RP ~= 5.77
14 day is 1846 RP. 1 RP ~= 7.58
Finally we have something that is worth more. Now, if you play for 24 hours straight, at ~30 minutes a game, ~6 minutes between end of last game and start of next game, and we assume you do everything outside of the game inside those 4 minutes (bathroom, eating, smoking, w/e).. that's 40 games in 1 day at an average of 100 IP per game: 1 RP ~= 13.8 IP. Now it's way better.
Personally I buy the 3 day boost when I know I have 3 days where I can game 15-20 games a day comfortably. I know I don't have the time for 2 weeks straight of 10 games a day, and I don't usually go on 24 hr binge-gaming sessions. If you don't have time for 10+ games a day, you're better off using your RP for buying the expensive champs, and using the IP you get in game for runes or cheaper champs.
Best Answer
No. You get a chest for receiving S grades when you have avaiable chests to be received, and are not currently considered with negative behaviour (Recent Restrictions/Bans).
That means that once you are ellegible to receive chests again you will also have to get S ranks to obtain the chests. You can't obtain S ranks, and once your punishment expires obtain the chests.
The same is valid for when you're out of chests to be gained. You can only gain another chest when you get one avaiable (You get one new available per week), even if you get S ranks.