If you have filed another stat dec, they
cannot sell the vehicle or take further action till your case is re looked at by Northampton.
If they refute your reasons, then you will be given another opportunity to have the matter heard with an N244 which you will receive with the notice from Northampton and information about your stat dec.
Information is here -
TEC
Your solicitor is correct in that it is advisable if you need your vehicle back, to pay and then fight back for a refund on any over charges or to dispute the process.
Yes they can seize property on private land.
Yes the warrant does give them the right to seize goods to pay the debt, however they should follow protocol and send you a first letter and at least attempt to make contact via visits.
The letter is charged at £11.20, the first visit and second visit are also chargeable, however if you believe none of these happened, then you can file under Section 7 of the Data act for a full breakdown and information about fees.
If anything is in dispute for fees visits charges etc, and the Section 7 highlights this, you can file against them with Money Claim online which is a simple process and costs £35.
It will not proceed to court if you have not followed the process of gaining the Section 7 and outlining your issues and request for monies to be returned before filing.