Bidding lower then the genuine price per the market is never appreciated and recommended. But pressuring your client to increase the budget is also a suicidal approach when making a proposal. The budget of the project will always be pre specified. If you are not comfortable with the budget, don't bid. Find other projects with higher budget and bidding on them would be a butter idea. Always bid on the projects which are matching with 'your expectations. It wouldn't only increase your conversions but also help others who are comfortable working in the existing budget.
Lack of patience
Making a proposal and sending chat messages wouldn't help you get the project. The client still have to go through your proposal in details and also checkout other proposals. It is a time consuming process. So it is better to wait for response after making a proposal. Ideally there is no time specified to follow up but its good to wait for 24-48 hrs.
Being too personal too soon
Personal questions too soon are a big turn off. In your proposals avoid asking personal questions which are not related to the project. Some clients might be comfortable answering them but others would not, but why take a risk? Wait for a right time to develop this rapport.
Pushing and bragging too much
The 'Best in market' speech wouldn't help you every time. Although, it is important to let them know why you are a good fit but representing yourself as the only hope wouldn't be fruitful either. And asking them to give you the project becomes pushy. Avoid this bragging and pushing. Give examples of your past work and represent yourself as a good fit but not the 'only fit'.
We all have gone through that phase when getting a client felt like chasing a white whale. We invest our time and efforts in impressing a new potential client and still end up getting no reply. Most of the times, the reason behind it, we approach them in a way that is easily ignorable. I am sure if you would have been in their shoes, you would too reject your own proposal.