Hello Tephrochr,
Your Cymbidium is quite a different orchid from the Phal. Cymbidium are more hardy,and tend to need more water than many other types of orchids.
You have made a great start with cleaning up the foliage. An insecticide is a good option for dealing with your pests.
You have mentioned that its currently in bloom; general advice is that its not a great idea to repot whilst in bloom, as for sure it will put the plant under stress and bring your blooms to an end quicker than if you had not repotted. Ideally, repotting should be carried out during Spring (after flowering). If you repot at other times of the year you may damage the new emerging flowering shoots (see left hand side photo; its a tighter shoot compared to a leaf shoot) and avoid damaging emerging leaf shoots (see right hand photo; leaf shoots are more open and have emerging leaves).
Also, yellowing leaves which you mentioned can be a common occurence and can be removed
Here's a picture of what our discarded pseudobulb/roots looked liked when we repotted a while back!
When you do repot, Use a moisture retaining, firm orchid compost. When topping up with compost the base of the pseudobulb should be at or just above the compost level.
Silkens advice is great - keeping the leaves moist will deter spidermites.
As for secret tips, we have always had great success with a tomato fertilizer (from mid-summer to early spring) in place of an orchid fertilizer. It's high in phosphate and potash, and low in nitrogen. You'll find your cymbidium may need watering every day in the hot summer months. In the winter, water them about every 10 days. Always water in the morning as this will allow the compost time to drain before the colder night temperatures. We feed on alternate waterings throughout the year, but we use an orchid fertilizer with a high Nitrogen content in spring and early summer to promote good leaf growth. As just mentioned, from mid summer, we change to the tomato fertilizer. This change tends to promote flowering. We continue with this fertilizer until the following spring. Due to the watering requirements, ensure your pot has plenty of drainage holes to disperse the excess water. Also, do make sure there is a gap between the bottom of the pot and the ground to allow excess water to fully drain away.
It is advisable to place your Cymbidium outside in summer, preferably not in direct sunlight. In summer the preferred daytime temperature is around 86F, 30C, falling at night to around 59F,15C. Most Cymbidium, but not all, require a distinct drop between night and day temperatures to initiate flowering. When its outside, protect it from wind and heavy rain (which will damage the leaves). A rainy summer could result in your cymbidium having few or no flowers later that year. All year round they need humidity. A sprinkling of tepid water over the leaves in the mornings will help to achieve this.
In the autumn, before frost damages your Cymbidium, place it in a conservatory if you have one, giving them maximum light (or they will not flower during Autumn / Winter). Temperatures during Autumn/Winter should not reach below 8C, 46F
Good luck, and let us know what you decide to do