I have to rest from exercise at least once a week. Maybe twice, but hope to make it just once a week.
Can we do dynamic / static stretch those days. Especially if we have have joint stiffness, i.e hip stiffness.
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Short answer would be that light work (walking / light jog, other low intensity work) and stretching on your "rest" days will help your recover from intense workouts. You should actually look for ways to be more active on off days instead of avoiding any work. Make sure you stretch after warming up to avoid injury (contrary to what many people do - stretching before warming up or as a warm up).
However I feel you might have a misguided understanding of the "need" to rest. If you feel like training more, you could (and probably should) train more. Overtraining does not happen as easy as some people try to make it and definitely has not happened if you feel like doing more. In fact you should avoid doing very intense workouts more than 3 days in a row but very few people do such taxing workouts every time anyway.
Then you say you'd try to have only 1-2 rest days per week. I would advise you to make your goals to be that you train in a way that lets you progress in a measurable way. Training more often will at certain point help you achieve that. But as an example - if your workouts consist of running, you could run more total distance per week, or run longer distance same time or run faster the same distance or combination of those. For weight training it could mean to lift more total amount of weight (sets x reps x weight) on your exercises or do same work in less time or combination of both.
It's enough that you start somewhere - let's say 3-4 times a week and make sure to progress based on your goals. At some point you might feel you need to start working 5 or 6 days a week (and maybe more frequent shorter workouts will feel better), maybe some days purposefully lighter, with some stretching. But again the goal should be to progress, not to train every day for the sake of it. If you start by working out (almost) every day driven by high motivation you might soon burn out as CNS (central nervous system) fatigue accumulates.