Varanamayiram composed by Kodai Nachiyar is a classic one. What are the benefits a devotee can obtain by reciting this? Can selected slokas be recited every day?Answers not from wikipedia or divyadesam. Purana reference is must.
In general, you won't find Puranic references to Andal ormthe Alwars except in the forms of prophecy, since most of the Alwars were born after the Puranas were written. So if you want to know the benefits of reciting a particular Alwar pasuram, your best bet is to turn to the works of Sri Vaishnava Acharyas, not the Puranas. Now thankfully the first Sri Vaishnava Acharya Nathamuni, who as I discussed in this question compiled the Alwars' poems into the Naalayira Divya Prabandham, composed a Phala Shruti for each poem outlining the benefits of reciting it. Here is the Phala Shruti for Andal's Vaaranam Aayiram poem:
AyanukkAha tAn kaNDa kanAvinai
vEyar puhazh villiputtoor kOn kOdai Sol |
tooya tamizh mAlai eeraindum vallavar
vAyu nan makkaLai peTTru mahizhvarE ||
This decad of pure Tamil verses by famous Villiputtur-king’s daughter Goda, describes her dream of marrying the cowherd-Lord. Those who sing it will be blest with good progeny.
That is why this poem is recited so often in weddings.
But that's not the only benefit of reciting it. Ultimately the purpose of reciting the Alwars' poems is that they encourage devotion to Vishnu and thereby can lead a person to being united with the lotus feet of Sriman Narayana.
Let me also quote a Taniyan or invocatory verse which the first Sri Vaishnava Acharya of the Vanamamalai Matham (the Sri Vaishnava Matham I belong to) composed about the Nachiyar Thirumozhi, the collection of Andal's poems that Vaaranam Aayiram is a part of:
kOlasuri Sangai mAyan sevvAyin guNam vinavum
seelattinaL ten tirumalli nAdi, sezhunguzhal mel
mAlait toDai ten arangarukkeeyum madippuDaiya
sOlaikkiLi, avaL tooya naRpAdam tuNai namakke
This taniyan states emphatically that only the sacred feet of ANDAL, whose sweet words resemble those of a garden parrot, and who brought greatness to the vEyar kulam of periyAzhwAr, are the refuge for us (today, tomorrow and for ever).
In any case, to answer your other question, yes, you can recite any Alwar Pasuram at any time; they don't have caste restrictions or other restrictions unlike the Vedas.