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There is probably a more elegant way, but how about

Number of lags here means to use many lags, For example 3 would mean using x(-1) x(-2) and x(-3)

If you estimate a least squares equation, the equation remembers the entry in the equation specification field when you hit the Estimate button. Switchreg doesn't remember the entry.

You are asking people to do your work for you. The best way to get help is to show what you've already done and explain where you are stuck.

Here's a snippet of a switching regression, simple switching on postwar GDP growth. Note that the standard error of the regression is larger than the standard deviation of the dependent variable. Is that odd?

lagged inflation is just inflation(-1), no d().

I believe that Excel takes logs.

Yes, the problem is you have more coefficients than variables.

It looks like you are trying to estimate four coefficients with only three right=hand side variables.

Remember that regression equations include error terms. The error terms here are not the same.

If data has a unit root, then it can be problematic in a regression. Data which is trend stationary is not generally a problem. So you shouldn't need to detrend at all.

I'm not sure why you want to detrend the series, but you could try regressing on both a trend and a trend squared,.

gets

If the last in sample probability is last_prob and the transition probabilities are p11 and p01, then I think you want something like

prob(T+1) = last_prob*p11 + (1-last_prob)*p01
prob(T+2) = prob(T+1)*p11 + (1-prob(T+1))*p01

etc

Have you tried Proc Make Regime Probabilities?