I am NO expert.
I disagree some with the view that no one can say what the chances are. Sure, no one can reliably calculate a precise probability. And it is worth emphasizing the actual outcome is not determined by chance . . . what matters are the specific details, the facts and circumstances in the individual case; these are what will most influence the outcome. For the PR who remains in Canada, how long before the appeal is actually heard by the IAD can be a significant, potentially positive factor, but absent other substantially favourable H&C factors that has not, historically, tipped the scales much.
Which means, sorry to say, frankly it appears that the odds are not good, ranging from poor to very poor. Poor enough it would be prudent to anticipate that the appeal will be dismissed and the decision to terminate PR status thus enforced. Sorry to be the bearer of bad news. For most the situation warrants being realistic. Some, perhaps many can afford the gamble, taking their best shot. And I do not mean to discourage doing that. Many others, however, have other priorities to consider (costs, family separation, among other personal consideration) and will want to approach their decision-making pragmatically.
Which leads to questions about the next step and how long the appeal will take.
Key next step: see a qualified, reputable immigration LAWYER. A licensed immigration lawyer with experience in these kinds of cases.
NOT a consultant. If cost is a concern, start with a consultation, a paid for consultation (free consultations are rarely if ever worth any more than what you pay for). At the least, a lawyer can offer a more realistic, practical assessment of the situation, and very specifically describe the process going forward. If full out representation is affordable, that can improve the odds of success (cannot guarantee success, and in this situation even with a lawyer odds remain not so good).
But, that in turns leads to the other part of the question, the HOW LONG question.
How Long Will An Appeal Take?
Appeals, like everything else related to immigration, are currently badly delayed due to the pandemic. Historically they have ranged from a little less than a year to well over a year, more than a few approaching two years. The latter seems quite likely now.
My sense is that you have conflicting preferences. If a favourable outcome was more likely, you probably would prefer a faster process, the sooner the better.
However, subject to what a qualified lawyer says, the longer it takes to be heard by the IAD, the better the odds of a successful, favourable outcome. But that is without any assurance how much this can improve the odds. This leads back to the importance (if pursuing the appeal and keeping PR status is a high priority) of obtaining a lawyer's assistance. For information. And for navigating the process in the most advantageous way, which might be to encourage some delays (with the objective of increasing the PR's establishment in Canada as a positive factor in the appeal), as well as presenting the case in its most favourable light.
Also note: a lawyer can better review the whole procedure to see if there was some procedural error in the adjudication procedure resulting in the 44(1) Report and Removal Order, that might render that not valid in law (a technical reason for allowing the appeal). This is NOT likely. Not at all likely. But it happens. In those isolated instances it happens, it can save the PR's status.
Other Observations:
I will mostly defer to others in regards to most SIN questions; except to say, concurring with
@scylla, that as a PR she can legally work in Canada. How the situation will practically affect obtaining employment, dealing with CRA, and such, there appears to be a good deal of uncertainty.
I will push a little harder than
@armoured in distinguishing your older son's experience. While a different outcome was possible, perhaps more likely, if he had applied for a PR Travel Document based on H&C reasons (given the minimal stay in Canada initially), the
removed-as-a-minor factor (whether addressed explicitly or simply an understood factor given his age upon arrival) undoubtedly tipped the scales favourably in however things went (whether border officials waived him through, authorizing entry without adjudicating RO compliance; or adjudicating RO compliance and determining he would be allowed to retain status based on H&C reasons, which would more definitively have been based on the
removed-as-a-minor factor). But, as
@armoured notes, your spouse does not have that H&C factor and does not appear to have a good H&C case otherwise.
Overall:
Best case scenario almost certainly means being separated for a long while, YEARS, during which the outcome will remain in limbo, with likely affirmation of the decision to terminate PR status looming overhead. . . . That is UNLESS you are eligible to apply for and obtain PR status again (noting eligibility qualifications have changed a lot since 2004).
If there is a fair chance of qualifying for and obtaining PR status again, you could submit the forms for renouncing PR status, or apply for a PR TD which when denied will terminate your PR status, and then begin the process for applying for PR status anew, and be doing this while waiting for the outcome in your spouse's case.
