Hi Maj
Sorry for only replying now but just saw the message.
Let's first talk about expressing
There is definitely no harm in introducing an occasional bottle now & then. What you express is absolutely no indication of your supply. Remember, your baby is the best in getting a lot of milk out of you. No pump can get close to that

Every woman is different on how their breast reacts to a pump. A bf baby will drink anything between 60 - 180mls. Also google paced bottle feeding to ensure that she doesn't overfeed. My son only drank between 60-90mls per feed when I went back to work. Also, the teat of the bottle will never increase cause your nipples stay the same size throughout hahaha
You can maybe directly after a feed & also express on one side while she feeds on the other (that's only if she only feeds off one breast per feed).
Now for the tugging at the breast:
Does she do it at a certain time of day?
Does she cry or moan when doing it?
Tugging at the nipple at that age is quite common though, but can also be an indication of a slow flow. Does she only suck or can you hear her swallow? If she is swallowing constantly then your supply is fine.
How many wet & dirty nappies is she having a day? That is also an indication of a good or bad supply.
Most babies go through growth spurts during the first few days at home and around 7-10 days, 2-3 weeks, 4-6 weeks, 3 months, 4 months, 6 months, 9 months, etc. Many babies are fussy during growth spurts but growth spurts can also happen any time.
Babies who are starting to notice the world around them can be notoriously distractible. Any kind of new developmental step that baby is working on can affect nursing temporarily, whether it be fussy nursing behavior or simply more frequent nursing.
Your baby is feeding every 2hrs, which is good for a bf baby as they feed more often than a formula fed baby. As long as she has free access to the breast, you don't really have to worry. Just make sure she latches properly & has a deep latch.
If baby is fussy right when your milk is letting down (or immediately after), there’s a good chance that the fussy nursing is related to a fast let-down. If baby is fussy before let-down, or a few minutes into nursing (and a while after let-down), then baby may be impatient for the fast flow of milk that comes with let-down. Fussing at the end of a nursing session (or what seems to be the end) may mean that baby needs to burp, or is ready to finish nursing, or just wants to suck (and doesn’t want to deal with a new let-down at this point), or wants to continue nursing on the other side or with a faster flow of milk.
Also check if baby might have thrush inside the mouth. Inside of the cheeks & lips.
Here are some links to read up on & see if anything stands out.
Also, try to burp her once or twice during her feed & immediately afterwards to decrease the throwing up a bit.
The best way to increase your supply is to demand feed as often as possible. Making sure she has a deep latch, drink enough fluids, cut down on stress and a lot of skin to skin. If she is picking up & growing in length & head circumference than all is well with your supply. If you see she is dropping weight etc then you can get Prolak at dischem. It's also homeopathic.
Please shout if there is anything else!!!!
kellymom.com/bf/got-milk/supply-worries/letdown/
kellymom.com/hot-topics/reflux/
kellymom.com/hot-topics/low-supply/
www.breastfeedinginc.ca/content.php?pagename=doc-BC