There is no increase in autism directly related to vaccines. Of course it's possible that some ingredient in vaccines along with thousands of other things we come into contact with every day is increasing the levels of autism but since you can't avoid all of it, it's not a good reason to avoid vaccines considering that they do offer such an enormous benefit. The levels of mercury, thiomersol etc are many many times lower than the amount considered to cause a risk. If you're worried about environmental contaminants it would make more sense to put your efforts into avoiding things which are less beneficial to your child's health such as plastics, processed food, pesticides, flame retardants etc.
Your baby sounds like a totally normal 4 month old to me - I have a 4 month old too and he is my second and he is very different to my first, so some differences between babies can also be their personalities. Do you know many other babies the same age? Of course this may increase anxiety if there are particular things you're worried about but it could also calm your fears if you're able to see that babies even of this age have huge differences in their abilities and preferences.
For example, my first baby was very determined to move (I just looked up my old diary today, and he was rolling already, whereas his brother is not), and he used to react with total delight when I kissed him whereas DS2 doesn't really react to kisses at all. But DS2 is very vocal whereas DS1 was not at all vocal until he was older. I get lots of intense long eye contact from DS2, I don't remember if I did from DS1 - but the lack of this can absolutely be personality, not a sign of autism. DS2 is very intent and concentrated on things and tends to study things quite hard and seems to have an air of taking it all in and working it all out whereas DS1 wanted to get into things and do them and has always loved social contact/affection to the point of being quite clingy.
Interestingly DS1 has ADHD, but it's pretty mild (it's extremely similar to mine, which was not picked up as a child at all.)
To go through your points:
How regularly would you expect eye contact? Your expectations may be off. I'd say it's fairly normal for them to want to look at other things. When younger than this they can't focus on faces very well and tend to look at abstract but high contrast patterns. By 3-4 months they start to see faces better and pay them more attention, but other things are still interested. They won't be glued to you all the time.
It is WAY too early for peekaboo and babbling. Totally normal for them not to react to this yet and babbling is when they start making consonant sounds and IIRC is more like 8 months, not 4.
Rolling isn't expected until 5 months, if it happens before it's early, so you're fine there. It's totally common for them to want to be held on their feet, if tiresome. Both of mine have liked this. DS1 even used to try and stand in a sling!
To me it would be strange for a young baby to be still, that's not very natural for them aside from perhaps when they are asleep (and even then a lot of them are twitchy/wiggly).
Repetitive movements at this age are her learning how to manipulate her arms, hands, objects. This is absolutely normal and actually a really good sign because it shows that she's practising things. DS2 does this a LOT and we'll often notice that for a few days at a time he'll repeat a certain action and then a couple of weeks later, it's perfect and he can do it without thinking.
I have to run as he's crying but will continue on phone :)