1) i'd put bitmixer on top of them all because it is the oldest mixing service around (started in 2014-1-10 on the forum)
2) any online exchange or gambling site can be used for this as long as they are legit places and you are comfortable with using them. and the only benefit is that this method is super cheap but the downside is that they will keep logs. so it is only good for regular people who don't mind that.
3) this method is a bit complicated and i have never personally done it but it is done using a second coin (Monero) you change bitcoin to XMR and this altcoin offers anonymity and then (there may be an step here) you change it back to bitcoin. you can use shapeshift or any exchange for this.
1. The table is in alphabetical order, and, in general, the question which service is the best and was opened earlier is highly controversial.
2. At first glance, it seems so. However, I would like someone to share a real experience with concrete examples. The devil is in the details, as you know. After registration some sites may require the scans of documents or check the phone number. Some of them can do something similar when smb withdraws their money. Others may delay the payment for a long time. And it does not suit to us. In addition, I would say just the opposite - the less legitimate site is the more suitable for the mix it is, I mean that there are low chances that site will share logs or has logs at all. i.e the less the site is legitimate, the more it will be similar to the site of the first group.
3. Yes. Mix through altcoins is a third way, which I had not organize and describe here yet. (I am engaged in this now). This way has some advantages (my thoughts about them I will write later) and a some minuses. One of them is that this way is, in fact, too complicated and you need to press "a lot of buttons."
In your case, you must register monero purse. Load btc to exchanger. Exchange btc to monero and then move them back into Bitcoin. As for the fees, they are quite acceptable. I looked through the shapeshift
Instant Rate 1 BTC = 96.4909903 XMR
Instant Rate 1 XMR = 0.01020749 BTC
Miner Fee: 0.02 XMR
That is for 1 btc the output we get: 0.9849. So the commission will be 1.5%. It`s acceptable, isn`t it? However, the extremely small size of the market spoils everything.