Yes you can. I have 4 current accounts. I used to have one with Santander and 2 others but I cancelled the Santander one when they started charging me for not paying in enough each month.
I have current acounts wth three banks (including Santander) and all pay me better interest than I can get in a savings account or bond.
You can get round the deposit restriction by setting up "round robin" payments. (Pay £500 from Bank A to Bank B on 1st of the month; pay £500 from Bank B to Bank C on 3rd of the month; pay £500 from Bank C to Bank A on 5th of the month). The bank's computer does not care where it comes from or goes.
Yes, of course you can . Many years ago banks used to frown upon customers who had accounts in more than one bank. Santander have a good deal going with their 123 account which , if you meet the criteria, give a better return than some ISAs