I definitely think it is largely an adjustment phase as well as a developmental one. She's going through a massive upheaval and is quite probably feeling very unsure of herself, and her place in the family. Especially if the times when your DP is around usually result in her being ignored or excluded - heck even as adults we'd be pretty put out by that, let alone a child accustomed to being the centre of attention. Can your DP come join in with activities you're doing with her like the crafts etc, or if you're watching TV or a movie together she can come snuggle with both of you? Have a few fun family events planned where you all get to spend time together as well as times where each kid gets one on one time with your DP and with yourself, it will help a lot in giving her confidence in getting to know your DP as well as lessen the feeling that he's taking you away from her rather than being someone else to think she's awesome.

With the lying - keep consistent. You and your partner really need to be on the same page as to what's acceptable behaviour, and what's not as well as what the consequences will be. At 4-5 years old, it needs to be firm and definite - so having to go get her barbie computer and hand it over for a set period of time, or not being allowed to watch TV for x number of days rather than the more undefined "till your behaviour changes". And stick to it - she will cry and carry on to try get you to back down, but I found a simple "I'm sorry, but you have lost it till Friday because you lied about x" as a quick reminder that you mean what you say works wonders to make her stop and think maybe this lying bizzo isn't all it's cracked up to be.

Big hugs babe, blended families are a tough gig at the best of times and everyone will require a period of settling in which may well have a few of these kind of "teething problems". But keep going, it does get better eventually