It may be sometimes dangerous to do it - I think it should not be a usual pattern. In general, it's preferable to outflank opponent from its front-hand side.
While doing a jab, you are commonly doing it in the same moment as stepping with front leg - and weighting it for a moment. You may change this move - and not only do a step, but turn your body - and do a side-step - as you described it - with circulating on the moment-weighted front-leg.
In both moves, you are weighting your front leg for a moment - and it is too easy (but without side-step - far less easy - your jab is faster) to counter you with a right(assuming, that your front hand is left) lowkick. Jabs are usually light, so your opponent surely prefer to block or absorb it, and make a lowkick to your weighted leg. And when doing a side-step such lowkick may check your balance - because it will interrupt your move.
Side-steps are good when you catched your opponent's middle or high-kick move, and you are side-stepping to the opposite side - you evading kick and flanking your opponent.
As variant, you may try to do a right(asuming your opponent front hand is left) side-step and do a right hook to counter his jab. But I think, that lowkick is preferable - you are spending less energy and, I think, you have better chances to land a hit - it is more fast to do. And after a successfull right lowkick, you may continue with left middlekick, shortening the distance.
About crosses. When you are crossing - if I'm understanding terms correctly - you are doing high amplitude punch - and you are weighting opposite leg, sometimes after a step with it. If you are feeling to step to another side - than you are doing a side-step, not a cross.
But maybe I misunderstood your second question. Please, note me, if it is so.