These are probably pretty obvious but:
Gigi - Georgina, Georgia, Gianna, Genevieve
Lex - Alexa, Alexis, Alexandra
Juni - June, Juniper
idk about your second question but I've always found it a weird concept. To take a random name, there must be hundreds of thousands of babies/children called Ava out there. None of those children look the same so how can anyone look like an Ava? Plus most babies look kind of similar to each other anyway.
Other ideas on little nicknames that connect to full names:
Ally - Alyssa, Alison, Alexandra
Callie - Clarissa, Calypso, Calliope, Calista
Cece - Cecily, Cecilia
Dottie - Dorothy
Etta - Henrietta, Minette
Effie - Elizabeth, Stephanie, Euphemia
Fifi - Fiona, Ffion, Fionnuala
Fizzy - Felicity
Gwen - Gwendoline
Ginny - Virginia, Regina, Georgina
Hattie - Harriet
Hetty - Henrietta
Ibby - Isabel, Isabella
Immy - Imogen
Indi - India
Jess - Jessica, Jessamy
Kiki - Kristine, Kristina
Kat - Katherine. Katrina, Kathleen, Katie
Lena - Eleanor
Liv - Livia, Olivia, Lavinia, Olive
Mimi - Miranda, Miriam, Marie
Mel - Melissa, Melanie, Melisande
Minty - Araminta
Nikki - Nicole, Nicola, Nicolette, Nikita
Nat - Natalie, Natasha, Natalia
Pip - Philippa, Piper, Pippa
Romy - Romilly, Ramona
Ronnie - Veronica
Rory - Aurora, Lorelei
Saffy - Saffron, Serafina, Sapphire
Sephy - Stephanie, Perspehone
Stevie - Stephanie
Sam - Samantha, Samaris, Samya
Tilly - Matilda
Tavvy - Octavia
Tammy - Tamara, Tamsin
Tori - Victoria
Vivi - Vivian, Vivienne, Viviaane