1) Смотря кого придерживаешь (I mean, it depends on who you're "restraining"). We might use different words if you're talking about a 5-year-old child in a grocery store; or an adult who is just beginning to learn how to ski/ice-skate/scuba-dive (if you're the instructor) or an elderly person who has difficulty walking. With a small child, we might say "to keep on a short leash" (which literally refers to walking a dog!). With an adult, we might say "to escort" or "to assist with walking."
2) If you are making major structural changes (like knocking down an interior wall between two small rooms, to create one large room), we'd say "to remodel" or "to do remodeling." Superficial changes (like putting up wallpaper) would be "to redecorate." As a general term that covers both extremes, "to renovate" and "renovation" will work.
3) The "nose slime" (when it's wet) is colloquially called "snot." When the snot dries or partly dries into clumps, these are called "boogers" in US English, but I believe that "bogey" is the usual term in the UK. (Hence the "bogey-flavoured" jellybeans in Harry Potter.)
P.S. A детская горка is sometimes called "a sliding board", although Seraph's answer "a slide" seems to be the more common usage. And if the slide is spiral-shaped, kids invariably call it a "twisty slide."