You are asking about the “assumption” that the future will be like the past and, in that context, about justification and certainty, particularly the certainty of experience. You also ask whether the sceptical argument against induction is “not self-refuting”.
I’m not sure what you are getting at in the last question. The classical problem of induction is not self-refuting, if you mean by that a rational counter-argument with a sound conclusion or a clear demonstration that the sceptical argument is invalid. You are right that the sceptical conclusion is, in a sense, unacceptable, but that doesn’t mean that it is self-refuting; it's a reason for trying to find a solution.
It is worth pointing out that David Hume, who is always cited as the originator of the argument in modern philosophy, partly accepted the sceptical conclusion; he accepts that there is no relationship of necessity between cause and effect as propounded by earlier philosophers, particularly the scholastics, and I think that there is general acceptance of that.
The question of induction is another matter. Neither Hume nor anyone else has accepted the conclusion that formulating empirical generalizations on the basis of past experience and relying on them as predictions of the future should be abandoned. Yet no-one directly refutes the argument; the name of the game here is finding a way round it.
The assumption that the future will be like the past
The sceptical argument treats the principle of the uniformity of nature as an empirical generalization like any other (such as “my car always starts first time”) and then argues that since it is not necessary it must be uncertain. But I think that this is a gross over-simplification. We do not always abandon a well-established generalization as soon as a problematic case appears. We have many ways of dealing with it. When the evidence is slender, we are quick to abandon it; when the evidence is substantial (and varied), we have ways of adapting it to preserve past experience and incorporate new experience into a revised edition. We can revise the scope of the generalization (we thought that all large water-fowl were white, but now we see that only all swans are white) or we can create a new category (we could define swans as white and invent a new category for black swan-like birds). To put the point another way, “The future will be like the past” allows that it will not be the same as the past, and we are perfectly capable of adapting to new cases that are unlike the old.
Nonetheless, we do believe that a given cause (or set of causes) will produce the same effects wherever and whenever it occurs. There’s no empirical evidence sufficient to prove this; nor is there an a priori argument for it. Does it follow that it is uncertain or unjustified? It depends on what you mean by justification and by certainty.
In general, what counts as justification depends on the context, specifically on what kind of proposition you are talking about and the language-game it is embedded in. Ideas like this mean that one can respond to the sceptical argument about induction by pointing out that the sceptic is applying inappropriate standards to inductive conclusions. Again, the Uniformity Principle is plainly not a simple empirical generalization and in my view needs to be regarded in a quite different light. It is a requirement on specific causal laws; any hypothesis that doesn’t conform to it will not count as a causal hypothesis; in a sense, it is part of the definition of “cause” and “effect”. Perhaps more accurately, it is a methodological principle to be applied in formulating hypotheses and testing them. So it will be true because we will make it true and any justification will depend on whether we find it useful in practice.
There are (at least) two different (but related) meanings of the word “certain”. One is psychological and is used when someone feels certain. It is used of a state of mind and known not on the basis of evidence and argument but by introspection, so there is no doubt of the feeling of certainty; unfortunately, that is no guarantee of the truth of what one is certain about. (If I feel certain that it is raining, you cannot be certain that it is raining.) The other sense is objective and applicable standards depend on the context or language-game of the proposition you are talking about. It is a guarantee of truth, but is not attributed to any particular person. Naturally, there are sceptical arguments that this kind of certainty is never or rarely found in life and the reply is that the sceptic is applying inappropriate standards.
Can we be certain of our experience?
If you are talking about introspective experience, even sceptics have to accept that we are each a final authority on our experience, in the sense that we are aware of our own experience. But if you mean experience in the standard sense, in which we experience something when we perceive it and such experiences are a basis for claims about the objective world, there is normally room for a doubt, provided there is a possibility of resolving it. If there is nothing that could resolve the doubt, it doesn’t make sense.
Further reading:- Certainty, Epistemic Contextualism, Language game (philosophy)