ボクシングし始めたい is very colloquial and the correct is ボクシングを with particle. So, the different between them isn't し and を, but し始める(to start doing) and 始める(to start). However, ボクシングをし始めたい sounds unnatural, al least to me though without たい, two sentences sound natural. I hope this was helpful.

動く→動き始める(start moving) 走る→走り始める(start running) 料理する→料理し始める(start cooking) ...These are models of "verb+始める" style. し始める is just a case of these style. The verb which takes ~する form makes 〜し始める style. But other verbs also have "start ~ing" style in different ways. The reason why ボクシングし始める sounds strange is "ボクシング" is not a verb. We rarely say "ボクシングする" but say "ボクシングをする". Naked する should be replaced by another verb "始める" in such cases. Only in the case する is a part of peculiar verbs (感動する、興奮する、ホッとする…) し始める is used. There are other examples of "する+verb" style, but, not so many. しだす(start doing ← this style is more popular than し始める)、し終える(finish doing←but やり終える is commonly used. It's rare to hear)、しかける(almost do ~ing)etc...し食べるsounds utterly nonsense. (What do you want to mean using these phrase? To emphasize?)

ボクシングし始めたい→『ボクシングする』(verb) + 『始めたい』 ボクシングを始めたい→『ボクシング』(noun) +『を』+『始めたい』 Regarding this example, both mean the same, but the latter is more natural, I think. For example, 料理し始めたい sounds you want to start cooking now, on the other hand, 料理を始めたい sounds you want to do cooking in near future so, are planning to cook at home or go to cooking school, something like that.

Yes! you just answered my second question below! thank you! Do you know of any/many common examples of し＋verb in the fashion of し始める ? and how they might be used. し食べる？　し歩く？for example

し is a conjugated form of する (doing) so the English translation should be somewhat like this: ボクシングし始めたい I want to start doing boxing. ボクシングを始めたい I want to start boxing. I personally think を始めたい is more commonly used though. It's a safer option to use in a conversation maybe.. I'm not really good at explaining things but I hope my explanation helps hahaha!