So, is it possible to do all that without crossing the infamy limit of 15, or will I have to face a war of containment?
And how do I annex Burma, since it's in my sphere from the start of the game?
It is difficult, but not impossible. If you declare a war of conquest without justification, that's a infamy hit of 20. But you've got to work around that kind of infamy hit to keep under the limit. For example, since you don't need all of Panjab to enact the Empress of India decision, you can later release Kashmir for a -5 infamy. Johore, and other states that don't offer any other nation to release are a bit trickier. Either you rely on events and triggers to get CB against those nations or you bide your time: you want those asian nations annexed by 1890, for example. Without a CB, and using conquest against all of them as WG, if you wait for your infamy to reach zero before going to the next war, you will need roughly 53 years.
This estimate I make using the following logic: conquering Panjab = 20 infamy, minus 5 for releasing Kashmir; 20 for Johore and Burma; 4 for demanding Hong Kong as a concession from the chinese, plus 4 for adding them to UK SoI (when I did this, I was more with an eye towards Africa already, so I decided to SoI them diplomaticaly, but let's say you DW for it). That's 63 infamy; given that infamy lowers at 0.10 per month, that's 630 months before infamy get to zero again - or 52 years and 6 months. With some micromanagement, not wasting any time, it's possible to accomplish this in 55 years at the most. So, around 1887~1891 you will have conquered all those states and your infamy will be at 0.
Even so, those 4 years above the infamy limit, when it's lowering from 20 to 15, are tense; In my current game, I decided I would play UK and do all the historical conquests of the time. I got lucky a few times, since I managed to get CBs against Transvaal and Zulu, and managed to conquer them before 1880. But on the eve of the 20th century, I gave up on conquering Oranje and Egypt, since it would take too long to lower my infamy again. I just put them in my SoI and forgot about it.
There's other things you can do that lower your infamy, like the Webster-Ashburton Treaty (-2 infamy) or the treaty of London (-4 infamy), but most likely you will face at least a couple of wars of containment. You can, however, abide to the conditions imposed to you in a war of containment, since it reduces your infamy, but the penalties to your armed forces make it a bit risky.
EDIT: About Burma; I had forgot that they start in your SoI. I conquered them because they rebelled and got out of my SoI; I lowered my relations with them, to see if that would work to get someone out of your SoI - but that was the only case that worked, since I lowered relations with a few of the other indian minors, to no effect. You just need to get lucky sometimes in this game.