Unlike many household consumer items these days, your brass instrument is not disposable. Treated with good common sense, care, maintenance, and respect, it will last a long time.
Oil your valves frequently.
Every time you play would not be too much. Use 3-6 drops for each valve. Tip: Try oiling them after you play. Because of the remaining moisture in the horn from your breath, this will protect your valves and keep them lubricated until the next session.
Don’t even think about it! If you can’t loosen it with your hands, bring it to the nearest repair shop. If you try to free the mouthpiece without a tool that is specifically made for this job, you are guaranteed to scratch the mouthpiece or the leadpipe, or possibly break a solder joint, or even damage the entire horn. A trained service technician can take care of this in a few minutes while you wait.
Your brass instrument can benefit from a bath every couple of months if you take the proper steps.
For the bath, use warm (not hot!) water and dish soap. Dawn dish liquid seems to be traditional but any mild dish liquid will do. Avoid ones with heavy perfumes. Note: Hot water WILL remove lacquer. Keep it to being comfortable to the touch and you should be fine.
For a brass instrument with valves, remove and wash them separately ensuring that all felt washers stay dry. (See image for typical felts.) The felts are both on top of and under the valve cap.
If the washing container (tub, sink, etc) is porcelain or metal, use a rubber matt on the bottom. Remove all slides. Submerge the horn carefully. Swab all inside tubings with a snake designed for your instrument.
Drain the water and rinse all parts (except for the felts…keep those dry!). Let all the parts dry. Then re-grease the slides and oil your valves. If you use a trigger or ring on your 1st or 3rd valve for tuning while you play, use oil on those, instead of grease.
Even if you religiously give the horn a washing inside and out, there is natural plaque (calcification) that builds up inside the tubing from your breath. This is unavoidable. You cannot remove the calcification with just soap and water. This must be removed professionally with special tools and supplies. This should be done 1 to 2 times per year by a brass instrument service technician.