It seems like your baby has some­ small spots around their bottom area along with a red diape­r rash. This can happen when the diape­r stays wet and irritates their se­nsitive skin. Change diapers ofte­n to keep them dry. Use­ soft wipes and let the are­a air out before putting on a fresh diape­r. Also, try a mild diaper rash cream to soothe the­ irritation.

That redness on your nose might be­ leftover swelling. Don't worry though, with some­ gentle TLC, it should fade. Be­ sure to moisturize and use mild products. Staying out of harsh sunlight (and SPF!) will also ke­ep discoloration away. It may take time, but your skin will he­al. 

Pigmentation is not treated quickly. It takes a while for treatments to work. But you can visit a dermatologist and discuss this. Based on your particular condition, he may suggest some alternative treatments that may work for you like chemical peels, laser treatments, topical creams etc. Hope this helps. 

It is advised to see a dermatologist who will inspect the site and give a prescribed diagnosis. They may also give treatments options that can be used to lessen the lines visibility.
 

Those light pink or red patches on your baby's face which are also known as salmon patches are quite common and usually nothing for parents to panic about. They occur when small blood vessels are near the skin. No need for treatment as they usually disappear on their own by the time the child is 1 to 2 years. Simply keep the area clean and protect against direct sunlight. 

Excessive sweating, also known as hyperhidrosis, on the other hand, can arise from a large number of causes like anxiety, hormonal fluctuations even certain medications. Your thyroid and blood pressure readings are normal so we should move on to other causes like stress or diet. Keep your body cool, use breathable fabrics, and don't forget about relaxation techniques like deep breathing or yoga, and you will reduce sweating. If it gets worse, you should talk to the doctor about it first.

Itchy skin can cause these marks when scratched frequently irritating the area. To help them heal faster, try not to scratch them more; instead use mild lotions such as aloe vera. Additionally, ensure that the affected region is kept clean and dry always;  however, with time they will go away automatically if it takes long without any changes make a point of visiting a doctor for further assistance though you need to be patient since the healing process may vary from one person to another.