For acrylic anything, just wash and dry it in the dryer. That will even out the sts which is all that steaming will do too, and it probably needs to be washed anyway after making it to that length. Put it in a pillowcase or mesh bag to keep it from tangling up or stretching out in the washer. The steam created in the dryer will be enough to relax and set the stitches too, you don't have to use a steamer or iron to do that.
Generally the lace CO is a knit cast on. If that comes out too tight, go up a couple needle sizes just for casting on. The trick to keeping any CO loose/stretchy is to not pull the yarn tight either while making the new st or putting it on the needle. Just leave it alone and don't even tension it.
It's usually better to just switch the needle size. You can knit a size larger if your gauge divided into the stitches for that size will come out to the measurement you need. However, the fabric may be stiffer and denser than it should be for the item, so using another needle will retain the same feel the item was intended to have.
I would try both one size up and 2 sizes up on a sample. I've found the stitch gauge doesn't usually change noticeably with just 1 size change, but will for 2 sizes. You need a whole extra stitch per 4" so the larger size would likely be better, a 6mm should do it. Don't be worried about the needle size being so much larger, it's matching the gauge that counts.
It's okay to use a differnt needle than the designer used, they may be a looser or tighter knitter than you are - everyone knits with a different tension. As long as your sts aren't so tight they're difficult to move on the needle or work with, you're fine to go up in needle size. Best way to make a swatch is to CO about 6-8 more sts than given for the gauge and work about 8 more rows; you can also do a couple rows of garter stitch then use another needle on the same swatch. Count your sts across 4" and see how that compares. Then wash and dry it like you will the finished item and count the sts again to see if that changes anything. Acrylic yarns may not change though they can sometimes relax a little in the dryer. But that's what 'after blocking' means, just wash and dry is fine, you don't need to stretch and pin it out. It's very likely the gauge was measured on the finished item after blocking that's why you should see what the yarn does beforehand.
Now, why is it important? You're getting 5 sts per inch, instead of 4 - say the pattern is 30" around. At 4 sts per inch that's 120 sts, but the same sts at 5 spi would be 25" so that's quite a difference! On the one that might be only a little small, you can put the sts on scrap yarn to wash and dry it now to see if it relaxes. If it does, you're alright, if not, take it out and do over after determining if you need to use a needle one or two sizes larger.
Many people go up 2 or 3 needle sizes to match the gauge, so that's no reflection on your knitting skills. You might look at how you tension the yarn and form the sts though. If you wrap it around your fingers, you could drop one of the wraps; when you make the sts, make sure to push the end of the needle all the way through the st so you're not making the st on the very tip. That would be like using a needle 2 sizes too small, the new st needs to be on the straight part of the needle. Also, if you pull the yarn after you make the stitch to make it 'nice and even', don't - making the next one will tighten up the previous stitch, they get even when you wash the finished item.
Holding the yarn in the left hand isn't 'wrong', it's called continental style and many many people use it, you may have seen other do english or 'throwing' with the yarn in the right hand, but neither is right or wrong. There's also several other different styles too. The color blips showing at the change is normal, and I think if you did them on the other side, they'd show too, it's because they show on the purl bump.