The true sticklers for grammar will say no, because "besides" is a conjunction (or I guess more specifically a conjunctive adverb), and many believe that the rule is that sentences shouldn't begin with conjunctions.

However, colloquially, beginning a sentence with "besides" makes perfect sense: "He didn't go to the store because he was tired. Besides, it was raining." I would usually say "And besides, it was raining", but I think that is just a personal preference.

You can also begin a sentence with "beside", but it has a different meaning. "Beside" is a preposition of direction: "Beside the store was a toxic waste dump."

You can always say "beside that", which is basically a synonym for "besides". Joining the relative pronoun "that" with the preposition of direction "beside" makes a phrase that pretty much functions like the conjunctive adverb "besides": "He didn't go to the store because it was raining. And beside that, there was a toxic waste dump next door."

TL;DR: Yes to your first question, and yes to your second question (though the meaning is different).