In informal American English, you can drop "and" from the combination "go and do" ("Do" could be any word here). For example, I need to go get some eggs. Go play, kids! Can you go find Max? This doesn't work if you need a different form of "go." You cannot say "I went get" or "I went got," or "I had gone get" but you could (for emphasis) say "I did go get the mail like you asked!" In the past tenses, you would need to use "and": I went and got the mail. As far as I know, this is only acceptable in informal American (and Canadian!) English. A British English teacher could provide the answer for that context.

In general you need 'to' after go, to say where you are going, or what you are going to do. There are some exceptions to this eg go home, go back, go away, go through, go here, go there. In casual spoken English 'to' is sometimes omitted eg 'go do this', 'go check it', 'go have a look'.

The word order has tricked you. "Check" does not really follow "go" in your sentence. Rather, it follows "can". It is really "I can go" but with the word order switched to turn it into a question. The infinitive "go" follows the modal verb "can" because infinitives always follow modals.