justMOVE.
couple of things.
your gym doesn't sound ideal at all but you can make the best of what you've got.
your results will not be like others, but it will be better than what you currently have.
so.
Start the movements in your program above with just the bar.
if you get the required numbre of sets and reps, then the next session increase the weight.
lets say you did SLDL with just the bar and it was extremely easy, then next session add 10kg.
if you did the over head press with just the bar and it was easy bt not as easy as the SLDL, then only add 5kg to the bar for over head press next time.
for your diet.
vegetarian and no eggs makes it pretty hard and again, your gains will not be like other peoples.. but, yo ucan still do it.
Can you eat fish?
if so, stock up on tins of Tuna in olive oil and grill up fish when ever you can.
you say you can drink milk and currently have 300ml per day?
if it doesn't make you sick, drink LOTS.. work up to 1L per day, then more and more.
keep going till you can drink 2-3L per day.
this will give you the calories you need.
Also if you are not eating it, eat mushrooms.
Can you eat Peanut butter (actually, do you get peanut butter where you live??).. if so, eat lots.
start with a Jar each week.
1.) increase the weight you lift on each exercise every session.. be it by 2.5kg or 10kg.. just keep increasing.
2.) eat lots and LOTS of food. and drink all the milk you can get. Find any alternative you can for meat to get your protein.. you need lots.