Tones and Pinyin, Foreigners vs Natives
Non-Native Speaker:


  As your Chinese learning has progressed, do you still intentionally analyze the tones of words you learn by speaking (e.g. remember 拼音 has two first tones)? 


As my Chinese has progressed (now on HSK level 5), I have discovered that I still "analyze" tones of words by speaking to a certain degree, but at my level I am more concerned with if the word sounds right or wrong rather than analyzing. For example, if I am speaking Mandarin to a friend and I say, "I want to buy some handcrafted products" "我想买一些工艺品” (Wǒ xiǎng mǎi yīxiē gōngyìpǐn) BUT instead of "买"(mǎi-to buy) I accidentally say “卖” (mài-to sell), I will instantly have a weird feeling and notice that I said "to sell" instead of "to buy". In short, the higher level you become in Chinese the better you get at noticing if a word sounds correct or not and quickly correct yourself based on the "internalized" tones rather than analyzing everything you say. 


  Do you even think about things like tone sandhi (which you may have learned complicated rules for at some point in your studying)?


I personally try not to overthink about things like tone sandhi (tone-change rules) because in the past it causes more harm to my spoken Chinese than it helps. Many of these tone sandhi rules can be solved by listening more to Chinese. The principle here is, the more I listen, the more I'll be able to notice mistakes in tones just because of how it sounds and then correct them. If you want to hear a professional Mandarin speaker talk about tones, then here is an interview from one of the most famous and fluent Chinese-speaking foreigners 大山 (Dashan) discussing Mandarin tones in English. 
Here is a link 
The Most Famous Foreigner in China